Carnival Phantasm: This is the situation that Shiro and Shiki have when trying to please their respective harems. If they try to date all of them, the girls will find out and go nuclear on them. If they decided to date the main heroines (Saber and Arcueid), the other girls turn into yanderes and subject the boys to all sorts of torment. They just can't win.
After Eren is exposed as a Titan Shifter, Weilman gives him a choice at cannonpoint: admit he's a traitor or say he's human. Eren, naturally, says he's human, and Weilman just fires the cannon anyway, pointing out that he can't prove it. Armin finds a way to prove it.
Much, much later, an all-out war for Shiganshina is being waged and the human side is suffocating under threat after threat. You can either be inside the wall with the Armored Titan, or on top of or outside it, which is being barraged by the Beast Titan's missiles. It doesn't get better, as the Armored Titan is eventually replaced with the Colossal Titan, who on top of being virtually infallible, starts setting the city on fire. Then the Armored Titan gets back in the fight. Essentially, the heroes' only choice is to pick how they want to die, and eventually they are forced to do exactly that.
Episode 11 of Puella Magi Madoka Magica comes down to this. An incredibly powerful witch called Walpurgisnacht is coming, and Homura has three choices:
Try to stop it without Madoka's help, fail and watch it destroy the city;
Essentially flee by going back in time and starting over, but knowing it will be worse next time because Madoka's magical potential will be even higher, causing her eventual witch form to be even more powerful.
During the Thriller Bark arc, the Straw Hats minus Luffy are fighting Oars, a giant-giant zombie animated by their captain's shadow. At one point Moria, the one who took the shadow and planted into Oars, shows up... in a cockpit built into Oars' chest. Now the crew has a dilemma; If they defeat Moria, Luffy's shadow will leave Oars, but the only way to get to Moria is to defeat Oars! Of course, Zoro thinks the problem just got simplified - both targets are now in one place.
In Chapter 697, Donquixote Doflamingo of all people finds himself in one of these thanks to Trafalgar Law and the Straw Hats kidnapping one of his important subordinates, who he'll only get back if he resigns from the Seven Warlords of the Sea. If he accepts, he'll be back to being a regular pirate and the admirals will hunt him down. If he doesn't accept, he'll have to answer to his business associate Kaido, one of the Four Emperors. The situation causes the Perpetual Smilerto show rage for the first time. After his attempt to Take a Third Option is foiled thanks to Kuzan, he reluctantly chooses to comply with Law's demands. About fifteen chapters later, we find out that he DID take a third option after all; he had the World Government falsely report to the whole world that he resigned just to fool Law and the Straw Hats, demonstrating that he has connections in the highest possible places. The plot thickens...
In the Zou Arc Sanji finds out he has been forced into an arranged marriage with one of Big Mom's daughters, and he leaves without much of a fight because you can't refuse an invitation by Big Mom. As Pekoms later tells the other Straw Hat pirates, this is because these aren't so much of an invitation, as they are an order, because if you refuse you'll be mailed a "present" with the head of a friend or loved one inside for disrespecting Big Mom. There's no excuse either, because much later, we see a man who couldn't attend one of Big Mom's parties because he had to be at his mother's funeral, and despite having explained the situation to Big Mom in a letter, he still got sent the head of his father. Even if you do attend, there's a good chance you might get killed anyway, such as in the case of Sanji's Arranged Marriage, because it's actually a plot by Big Mom and her family to assassinate the Vinsmokes to gain political power.
Hunter × Hunter includes one of these as the protagonists are beginning their journey towards the Hunter Exams. As they walk through a shady alley, an old woman pops up and asks a couple different examples, often asking you to choose whether to help one person you care for or the other. Leorio eventually tries to attack her in frustration before it is explained that the only correct answer is that there IS no answer. Before they continue, she explains that all Hunters need to understand that sometimes, you must face issues that you can never get right.
Fist of the North Star has Jyuza of the Clouds being beaten to death by Raoh to get him to reveal the identity of the last Nanto General. He ultimately dies without telling Raoh, but Raoh gets the information anyway when he figures out that Jyuza would only keep silent unto death to protect Yuria.
Medaka Box ends with Medaka and Zenkichi having one last fight. If Zenkichi wins, Medaka has to accept his marriage proposal. If Medaka wins... Zenkichi has to accept her marriage proposal. A win-win situation in this case.
In X/1999, Kamui must choose between the Dragons of the Earth and destroy the Earth to cleanse it, or the Dragons of the Heaven and save the Earth to preserve humanity. That is, between "saving the earth" and "saving humanity." Not having much inclination one way or the other, he chooses the Dragons of Heaven, because his two friends (Fuuma and Kotori) are the only ones he really cares about, and these two would die if the Earth was destroyed. However, no matter what decision he made, Fuuma would become champion of the other set of dragons and Kotori would die, which iswhat happens. Leading to ANOTHER Morton's Fork as he has to decide between killing his remaining friend Fuuma to fulfill his purpose... or get killed by Fuuma instead. In the movie he chooses the Earth over Fuuma and kills him, in the TV series he chooses to let Fuuma kill him to deploy a Thanatos Gambit that ultimately saves Fuuma and the planet, and in the manga no one knows what will happen.
Just before the final match at the 23rd Tenkaichi Budokai in Dragon Ball, Piccolo has trapped Kami in a tiny jar and swallowed it. If Goku wants to release Kami, he'd need to rip Piccolo's body open to get it... but if Piccolo dies, so does Kami. Luckily, Piccolo himself provides a third solution - During the fight, he makes himself so big that Goku is able to leap down his throat and pull the jar out!
In Princess Tutu, Princess Crow holds the fragment of heart containing the prince's love captive, but suggests that it can pick between herself and Duck. Duck is faced with the choice between confessing her love to the prince (which would cause her to dissolve into a mote of light, making it impossible for her to return the fragment of the prince's heart to the prince, as she's the only one who can) or allowing Princess Crow to win. She takes a third option and shows her love for him by dancing.
The climax of the preliminaries in the Battle City arc is a deathmatch between Yugi and a mind-controlled Jounouchi. The match has been set up so that both players are chained to an anchor that will fall after a set time, dragging whoever's still attached to it into the sea and causing them to drown. The only way to win is to defeat the other person first, in which case the winner gets a key that can unlock the chains. Either Yugi loses and dies, Yugi wins and his best friend Jounouchi dies, or Yugi runs out the clock and they both die. Eventually, they manage to force a draw, unlocking both keys- Yugi passes out and Jounouchi saves him. And as an extra measure, a mind-controlled Anzu was going to ingest a cyanide pill, or get crushed by a giant block, depending on if it's the manga or anime.
The Grand Finale of Battle City had something similar. Yami Marik had both Yugi and the true Marik's souls hostage; if Yami lost, Yugi would cease to exist (as would Yami, due to his bond with Yugi) while if Yami won, the same thing would happen to the true Marik. Even worse, Yami Marik does not have such a bond with his host, and will actually benefit if the true Marik is gone. While no third option is initially presented, one comes up when Rashid appears, giving the true Marik a way to fight back, and eventually regain control after Yami reduces his Life Points to only 1. Once in control, the true Marik surrenders, and since their roles have been switched, it's Yami Marik who ceases to exist.
In Duelist Kingdom, Yugi manages to use one Morton's Fork to counter an enemy's: faced with two doors guarded by the Paradox Brothers, and having determined that their Knights and Knaves puzzle was fake, Yugi still had to figure out which door led to the exit and which was a trap. Eventually he showed the brothers two coins, each apparently marked with one of the brothers' symbols, and said that, after closing his hands over both of them, he'd open one hand and whichever coin was in the closed hand was his choice. Revealing the symbol on one coin, the brothers indicated that he'd chosen wrongly- only for Yugi to open the other hand and reveal the same symbol, as he'd figured out that the brothers were just going to say he'd picked the wrong one no matter what, and thus he'd put a different symbol on each side of the second coin. Defeated, the brothers were forced to let Yugi and his group go (whereupon they discovered that both doors actually led to the exit all along).
In the first match of the Battle City Finals, Bakura tried to pull one of these on Yugi. His strategy involved using Dark Sanctuary, which can possess a monster, negate its attack, and damage Yugi's life points while restoring his own. He used The Dark Door in conjunction with this, limiting Yugi to only one attack per turn. But he also had Destiny Board, a card that after so many turns, creates an Instant-Win Condition, meaning Yugi couldn't just waste his turns not attacking, but couldn't risk using his only attack with a possessed monster. During the last turn, Bakura was one letter away from completing Destiny Board, but he also had the trap card Dark Spirit of the Silent on his field, which lets him negate an attack and force another monster to attack in its place, meaning Yugi would've lost from either Destiny Board's Instant-Win Condition, or from the effect of Dark Sanctuary. The only reason Yugi manages to win is because he manages to draw his God Card Slifer/Osiris, which the ghost of Dark Sanctuary couldn't possess.
Yu-Gi-Oh! VRAINS: In episode 108, no matter which combination of Ais cards GO could have destroyed after tributing Gouki Thunder Ogre to activate the effect of Gouki The Powerload Ogre, the end result would have been the same. Destroy Doshin @Ignister, Pikari @Ignister, and the set card, and Dark Templar @Ignister would soak up some of the damage while Ais Hymnthe set cardwould allow Ai to live with some life pointswhich is what happened in the anime. He also could have chained Ais Hymn to Powerload Ogres effect to tribute Dark Templar and gain 2300 life points before it could be destroyed and then use the graveyard effect of Ais Hymn to banish itself so Ai could survive the two direct attacks from Gouki The Master Ogre and Powerload Ogre. Attempt to destroy Doshin, Pikari, and Dark Templar, and Ai could have chained Ais Hymn to Powerload Ogres effect to tribute Dark Templar and gain 2300 life points before it could be destroyed and then use the graveyard effect of Ais Hymn to banish itself so Ai could survive the two direct attacks from The Master Ogre and Powerload Ogre, just like in the first scenario. Destroy Dark Knight, Doshin/Pikari, and the set card, and the remaining defense monster would soak up the damage from one of the attacks while Ai would still have life points remainingPLUS he could still have used the graveyard effect of Ais Hymn to gain some extra life points. OR he could have chained Ais Hymn to Powerload Ogres effect to tribute Dark Templar before it could be destroyed and gain 2300 life points, like in the first and second scenarios. Same result indeed; no way to win. Looks like Ai was lying when he said he would have lost if GO had instead destroyed Dark Templar.
And even though a Link Summoned Powerload Ogre is unaffected by other card effects, other effects can still be chained in response to it activating its effects. Since Ais Hymn did not even involve Powerload Ogreinstead tributing an @Ignister monster to gain life pointseven Powerload Ogres protection effect couldnt have stopped the inevitable loss GO had suffered.
In Anatolia Story, one arc has Nakia frame Yuri for poisoning Kail's eldest brother aka the King and sends the army to hunt her down. Her allies try to figure out how to save her, but their options boil down to (A) proving who the real murderer was (virtually impossible, since there's no evidence), (B) lying that one of them was the murderer (a guaranteed death sentence, considering that they'd be confessing to regicide), (C) Kail himself goes to rescue Yuri (which would pretty much ruin his vision of becoming King and saving the land, since he'd be foiling the arrest of someone accused of regicide), or (D) send word to Yuri to kill herself (which would prevent Nakia from sacrificing Yuri to curse Kail, as she plans to, but also would mean that Yuri ends up dead). The group is unable to decide what exactly to do, but the matter is resolved when one of Yuri's ladies in waiting, Ursula, willingly takes the second option: she confesses to the crime herself and allows herself to be imprisoned and executed, leaving Kail open to save Yuri without issue. Some time later, after Nakia's ally Urhi Shalma confesses to the King's murder (and is executed right there) and then Nakia is completely defeated, Ursula is officially exonerated.
In Assassination Classroom, this is how Takaoko intended to destroy Karasuma's reputation as a teacher for Class E. Karasuma had the option of either sending one of his under-trained students into a hopeless knife duel against Takaoko or refuse the challenge and let Takaoko remain as Class E's barbaric P.E. teacher. Karasuma takes a third option and sends the one student that has a horrifying potential in close-combat assassination. It works.
As a character, Shinji Ikari is an example across the continuities of Neon Genesis Evangelion and Rebuild of Evangelion. He starts out as an insecure, socially awkward Nice Guy who has trouble dealing with people, but also has plenty of untapped strength and potential. Once he becomes an EVA pilot, though, he experiences serious psychological trauma fighting the Angels. By the end, after Kaworu's death, he's become a self-loathing wreck, lost all hope and begs for others to help him. He's about ready to allow Third Impact and instrumentality to happen, just so he can stop dealing with all the misery and loneliness in his life. Rebuild has Shinji become much more assertive and powerful than his anime counterpart. What happens here? In trying to kill Zeruel, he ends up triggering Third Impact through his sheer determination, killing most of what was left of humanity and damaging the Earth's biosphere even more severely. Poor kid just can't catch a break.
The situation in Magical Girl Lyrical Nanoha A's was originally set up as "The WolkenritterbetrayHayate's trust and fill the Book of Darkness or she dies", though later episodes show that it didn't matter if they filled it or not. If they didn't fill it, Hayate's paralysis would spread to her vital organs and she'd die. If they did fill it, the Book would activate, kill Hayate, and blow up the planet (or Graham would freeze it and her in a pocket dimension for all eternity). It's only thanks to Nanoha and Fate's intervention that they were able to Take a Third Option.
The situation involving Dragon Shiryu's self-inflicted blindness in Saint Seiya was a consequence of one of these. He was fighting the very powerful Perseus Algol, who had the Gorgon Shield that let him transform people in stone statuesand had already petrified Seiya and Shun. Shiryu tried to close his eyes and fight that way, but it didn't work and his body began to petrify. Later he tried to blindfold himself, and it still didn't work and half his body was petrified (And in the anime, the Steel Saints were about to suffer the same destiny). So the only option he had left was plucking his own eyesand THEN finish the fight and kill Algol, which is what worked. So Seiya and Shiryu (and the Steel boys) were saved, but in exchange Shiryu was left blind from then on (temporarily in the anime, definitely in the manga).
The Irregular at Magic High School, after their Perfectly Arranged Marriage, Miyuki and Tatsuya's relationship reaches a stalemate because of this trope. The former constantly makes romantic advances towards the latter, yet also freezes up in embarrassment when he reciprocates those advances. Basically, no matter what anyone does, someone is going to feel uncomfortable about it.
Nou Come, the protagonist is offered absurd choices throughout most of the series, usually all of them are equally as bad.
In Bloom Into You, Touko, as a result of her self-worth issues, inflicts this on herself. While she's quite popular at school, she also convinces herself that the people who like her like her sister, by extension, since Touko has devoted her life to becoming her sister. Those who claim to like her for who she really is don't fare any better, since Touko hates herself, and wants to cast her old unremarkable self aside.
My Next Life as a Villainess: All Routes Lead to Doom!: When an ordinary Japanese high school girl is reincarnated into the life of an otome game villain, she realizes all her potential futures are like this. In the console game Fortune Lover, Katarina's fate is either exile (being stripped of her noble title and banished from the country as a criminal) or death (by either Gerald's sword or Keith's magic). Neither fate is appealing to Katarina, who does everything she can to Take a Third Option, such as building up her combat and magic skills for self-defense and learning agriculture to give herself skills to fall back on in the event of exile. She completely fails to realize that none of this is necessary; she's derailed the entire story simply by being a Nice Girl instead of a bitchy Spoiled Brat. It's to the point that when the event occurs where, in the game, Katarina was exiled or killed, all the characters who were supposed to be against her instead leap to her defense.
Chapter 60 of Goodnight Punpun summarizes Mama's situation with this: "Without a place to call home, she's spent her entire life running from herself and her loneliness. What a sad life it will have been if she dies in surgery tomorrow. And what a sad life it will be if she lives."
In Hino-san no Baka!Koguma ends up playing a game with Hino, in which Hino asks Koguma five questions about Koguma, offering an answer, and Koguma can tell her whether she's right or wrong (although Hino can confirm whether Koguma's response is true, when possible). If Koguma wins, Hino goes back to class, but if Hino wins, she gets to fondle Koguma's breasts. With the score tied, Hino's final question is to ask whether Koguma has A-cup size breasts.
Koguma: Wh-What happens if I say you got it right? Hino: I win, so I'll be fondling your boobs as a penalty game. Koguma: YOU'RE THE ONE WHO DECIDED ON THAT PENALTY GAME! Th...Then, what if I say you got it wrong? Hino: Well... I'll fondle them to check their size!! Koguma: SO THEN YOU'LL BE FONDLING MY BREASTS NO MATTER WHAT, YOU JERK!!