So there's these two characters, right? And they spend 95% of their time thinking about each other, what the other one's planning or doing, and whenever they converse their dialogue is incredibly emotionally charged. Fortunately, there's a nice family-friendly explanation: they're arch-enemies, right? Right...
This audience reaction trope happens when fans interpretsubtextual marks of sexual tension in a tense, antagonistic, or outright violent relationship, writing Fan Fic where it turns out the arch-enemies love each other. They might write plots that end in a Dating Catwoman resolution, or use The Power of Love to induce a Heel-Face Turn (or more rarely, induce aFace-Heel Turn), or exploit the existing dynamics to show a romantic relationship that's dysfunctional or outright abusive. If the antagonist's one-sided crush is already canonically admitted, in these Fan Fics the protagonist might realize that Stalking Is Love, and reciprocate the feelings.
Compare to Belligerent Sexual Tension and Foe Romance Subtext, where there's actual romantic tension between the characters.
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Anime & Manga
Pick a Shōnen protagonist. AnyShōnen protagonist. It's pretty much guaranteed that there will be tons of Slash Fic about him (or less frequently, her) and his (or her) arch-nemesis.
Attack on Titan fandom predictably has a huge amount of Eren/Annie, but there's also the less popular but still prominent Mikasa/Annie.
A popular Crack Pairing is Mami/Charlotte. As in, the witch that decapitated Mami and ate her body. How this came about is anyone's guess, but it got some support with Rebellion, where Charlotte/Bebe is Nagisa Momoe, who joins the cast.
Kill la Kill fandom had Ryuko/Satsuki, prior to the reveal that they're long-lost sisters. After that... it continued to have Ryuko/Satsuki, with a hefty dash of Incest Yay Shipping for flavor.
Durarara!! has Shizaya (Shizuo Heiwajima/Izaya Orihara) as its most popular pairing. The Homoerotic Subtext of their relationship is repeatedly lampshaded (especially in the anime) by other characters, most notably Yaoi Fangirl Erika, who is a prominent Shizaya shipper. Shizuo hates Izaya so much that he flies into a blind rage just by seeing him (and some times, just hearing his name), while Izaya espouses that he loves all humans — except for Shizuo, who he admits to being the only person in the world that he hates. The anime ups it a notch by having them so fixated on their rivalry that it mirrors the very disturbing romantic obsessions of other characters on the show.
Possibly the most popular ship in the Death Note fandom is Light/L. You know, as in the mass murderer and the detective he eventually murdered. Sometimes the fandom's reasons make little sense, but sometimes they have fairly good ones.
00/Silva shippers often complain that 00/Q became the Skyfall fandom's flagship, seeing as Silva aggressively came onto Bond multiple times throughout the film.
There Will Be Blood focuses a lot on this dynamic between Daniel Plainview and Eli Sunday, featuring several scenes of one beating the other up, physically or mentally. It's so prominent that nearly all of the fanfic written is for this ship.
There are fanfics of The Rocky Horror Picture Show that pair Riff Raff with Dr. Frank-N-Furter. This is possibly due to when Frank whips Riff... S&M anyone? Or due to this line (which seems to indicate that Riff was jealous of Frank):
Considering the whole nature of Rocky Horror, there's also fanfics that pair pretty much anyone with Frank (and has, according to various bits of dialogue and action, already paired himself with most of the major cast members at one time or another).
The weird thing about romantic fanfiction for The Dresden Files is that the Harry/Marcone fics make up a lot of it. This is weird because Harry is the compassionate, staunchly independent protagonist, and Marcone is the Badass Normal mob boss who controls Chicago. There are barelyanyhints in canon to build the ship on, and yet fics supporting it form a clear majority.
Sheriarty is one of the biggest ships in the Sherlock fandom, most likely due to the twisted and somewhat sexual (the line, "You've got to admit that's sexier," in context is a good example) fascination Sherlock Holmes and Jim Moriarty have for each other.
Given how much Foe YayHeroes has, there are a ton of Foe Yay shipping fanfics. Some of the most popular pairings seem to be Sylar/Peter, Sylar/Mohinder, and Sylar/Claire. It does make sense, because Sylar and Peter are the fandom bicycles of this series. The Sylar/Claire is actually somewhat canon; it's just one-sided on Sylar's part.
Although they eventually became friends, Quinn Fabray and Rachel Berry from Glee started off bitter rivals who acted quite sociopathically towards one another. Didn't stop Faberry slash fanfiction to be spewed out in their thousands online.
In Victorious, Jori is a popular ship consisting of two girls (Jade and Tori) who despise each other so much they've gone to psychotic lengths just to spite one another. And the ship is awesome.
Season 5 of Merlin seems to have produced the Merdred ship, thanks to a no-longer-a-kid Mordred and the show's famous copious amounts of Ho Yay. And that scene in 5x02 in which Merlin helps Mordred with his cloak.
Amongst Doctor Who fans, Doctor/Master has been one of the most popular pairings for decades. It long ago reached the point where the writers ship them as well, and ramp up the subtext to the point that it's barely subtext anymore.
Feuding wrestlers can often come across as this. However, a special mention must go to Bret Hart and Shawn Michaels, who's legendary feud went on for twelve years on-screen and off; since burying the hatchet in 2010 they now have regular Ho Yay with one another instead of the obsessive Foe Yay of yesteryear.
Randy Orton has this with almost all of his rivals over the years (no thanks to being a Launcher of a Thousand Ships), but none more so than with John Cena, easily the most prominent Arch-Enemy of his entire career (followed closely by Triple H). Orton and Cena do not like each other, and never have, even when they were both faces. When something puts them in even the slightest amount of conflict, it's enough to push them to the verge of feuding again, even though their rivalry was suppose to end (officially) back in 2009 after their infamous Iron Man match (which Cena won). Naturally, the animosity between them is enough for the fan girls to ship them together with a vengeance.
Chell/GLaDOS of Portal have a gargantuan following, what with GLaDOS continually acting like an abusive partner, though being the only characters made it largely inevitable. As of Portal 2 Chell/Wheatley can be this, depending on whether the specific fic writer thinks that the latter's actions were deliberate or a case of With Great Power Comes Great Insanity, and whether or not they depict Chell forgiving him.
Team Fortress 2 fandom loves pairing members of the opposing teams with each other, especially Sniper/Spy, who in canon hate each other with the fury of a thousand suns.
Miranda/Jack became a popular pairing despite (or rather because of) them being the most hostile to each other out of anyone in the cast, outright coming to blows at one point. As with everything else, the Citadel DLC for Mass Effect 3 gives this a Fandom Nod.
Ace Attorney has Phoenix/Edgeworth as the pairing of the fandom, despite them starting off as antagonists. Then Edgeworth slowly got a Heel-Face Turn and support remained strong, ultimately averting the trope.
Happens in-universe in Unwinders Tall Comics. Unwinder watches a few episodes of the anime Tokyo Delta Jetlag D and remarks, "I'd kind of like to see what would happen if Jaded and Colonel Gunarm hooked up."
Cecil, the narrator of Welcome to Night Vale, often gets carried away ranting about how much he hates every little thing about Steve Carlsberg. Some fans have assumed they're exes or that it's just Belligerent Sexual Tension. It helps that Cecil has a few canonical male love interests.
Made a little weird by the fact that Steve Carlsberg and Cecil are brother-in-laws, or he's at least the boyfriend of Cecil's sister
About 80% percent of RWBY fics ship Ruby/Weiss. And 80% of those fics are predicated on the assumption that Weiss' constant belittling of Ruby means they're in love.
Gargoyles fan-fiction does this a LOT, with one of the noteworthy examples being Elisa/Demona in many a Slash Fic. Where that particular pairing is concerned, approximately 50% of all fics dedicated to it usually portray Demona as the dominant and Eliza as the (initially) unwilling submissive, while the other 50% seems split between them either being very cool toward each other but gradually warming (no pun intended) or with Demona's human form coming into the mix somehow. There's also Goliath/Demona (it helps that they canonically used to be mates) and Fox/any female character.
Quite a lot of this for Jet/Zuko, who were briefly antagonists to each other.
Azula/Anyone, thanks to generating Foe Yay with a good chunk of the main cast. The most popular targets are Zuko, Aang, and Sokka.
This was how the Zuko/Katara ship started, until Zuko began to drift away from being an antagonist.
Sequel SeriesThe Legend of Korra follows in it's predecessor's footsteps, with Korra being paired with Amon, Tarrlok, and Tahno. However, most of the Korra/Tahno fanfics that take place after episode 6 are of the Hurt Comfort variety rather than Foe Yay for reasons that will become obvious after you see what happens to Tahno in episode 6.
Invader Zim fans love pairing Zim and Dib, usually revolving around their shared status as universally despised outcasts among their species. Much less popular is Zim/Tak and Dib/Tak, as while the pairings themselves are popular most of time in fics Tak just switches to the side of whomever she's boning.
There are a large number of Doug fans who pair up Doug and Roger.
There a lot of Superjail! fanfics pairing up the Warden with either the Mistress or Lord Stingray, both being his archenemies/rivals (though he remarks that he rarely gets to see the former). In the Mistress' case, there's the fact that she's a Distaff Counterpart and the warden of her own prison, and a cruel ice queen that rivals Warden's lighter-hearted megalomania. Thus, some fans insist she and the Warden must have feelings for each other and absolutely get together, although some stories seem to subject her to Chickification and have her personality altered to be submissive to the Warden (though some may insist on making her dominant too). With Lord Stingray, the fact that he tries to constantly one-up the Warden and that the Warden was initially impressed by him gives the Warden/Stingray fans quite the tension to go on as well.
Celestia/Discord is one of the most popular pairings that doesn't involve the mane six, and Discord/Twilight (as well as Discord/any of the mane six) pops up on a regular basis.
Luna/Sombra came about soon after the latter's debut, despite him being Killed Off for Real, as a foil to Celestia/Discord, and while it's nowhere near as common it pops up here and there.
Twilight/Trixie and Rainbow Dash/Gilda were very popular in the early days and support still goes strong years afterwards. Granted, the latter half of the pairings were more antagonists than outright villains, and Trixie also got a Heel-Face Turn.
Adventure Time fandom has a surprisingly large following for Fionna/Ice Queen, which could be the result of Ice Queen literally taking Fionna on a date. Finn/Ice King is a big no-no in the fandom however, with Finn looking and acting a lot younger than Fionna and Ice King being repulsive in every sense of the word.