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The Dick Van Dyke Show opened with the end of a particularly nasty marital argument. When Mary complains to her friend, she was being a pleasant wife and her husband was in an inexplicably nasty mood. When he complains to his coworkers, he came home to find her unusually lazy and nagging. The audience then gets to hear from the goldfish what actually happened: they'd both had bad days, and took it out on each other.
In an episode of Space Cases, when Catalina is asked to describe what happened with the Ion Storm, Harlan acts completely and utterly worthless and it's actually her who saves the day. When this flashback finishes, everyone says "...wait that's not what happened" and they ask for Harlan's version, which is...more or less the same thing but with Harlan presented as the hero and Catalina being useless and her obsession with Suzee being exaggerated.
An episode of Perfect Strangers have Larry, Balki, and their neighbor give differing stories to the police about an incident. Each version has the teller as the hero.
An episode of Happy Days had Fonzie, Chachi, Roger, & Potsie all giving differing versions of the same chain of events leading to Fonzie getting shot in the butt.
All in the Family had a Rashomon episode where an incident was seen from the points of view of all four principals - Edith's version was the objective, accurate one, of course.
Supernatural episode "Tall Tales" is a The Rashomon episode, with Sam and Dean telling their own version of the previous events to their Parental Substitute Bobby - and often end up arguing over who's telling the story and the exact details of what occurred. It is eventuality revealed that a Trickster (a minor god of chaos) has been messing with their relationship in order to distract them from the case at hand, so most of the narrative consists of whichever brother is speaking portraying himself as a suave, dedicated professional searching earnestly for the truth, while painting the other in decidedly uncomplimentary colors. In Sam's narration, Dean appears as a slutty, gluttonous pig with no standards, while Dean portrays Sam as a prissy, super-sensitive do-gooder with Camp Gay mannerisms. They end up working together to defeat the Trickster and sincerely apologizing for their behavior after closing the case.
The Farscape episode "The Ugly Truth" has four of the characters being successively interrogated about the destruction of an alien spacecraft by angry compatriots of the aliens who assume that any difference in the stories must be deliberate lies. While we can see that the characters are consciously or subconsciously framing events to make themselves look better, the central character Crichton finally delivers a Kirk Summation about how memory is fallible and no one person's description of something will ever be totally accurate. Notably, the aliens claim that this cannot be, as they always remember things in the same way.
"Scratch 'n Sniff" features Crichton relaying the events of why he had to leave a planet to Pilot. At several points, Pilot refuses to believe Crichton (even at one point suggesting that if Jool had lost as much fluid from her body as Crichton said, she'd be dead) and in the end it was left ambiguous how much of the story, if any, was true.
Leverage has "The Rashomon Job", in which each of the characters recounts how it was they who stole the golden dagger. In the end, Nate reveals the single true story and reveals who really stole the dagger. One running gag is everybody messing up Sophie's British accent. By Parker, she sounds like a dwarf from The Lord of the Rings.
The Black Donnellys: The narrator (Joey "Ice Cream") puts himself into the story in places where he couldn't have been, gets dates wrong by a year or so, and just has the general demeanor of not being a guy whose facts are ready to bank. On the flip side, the story he tells does not make him seem like a Marty Stu. He gets shut down by the ladies. He never plays a pivotal role in the events of the story. This leads us to believe we can accept at least some of what he is saying. Joey generally gives the sense of wishing he had brothers like the Donnellys, and that's why he inserts himself into the story, in a hopeful-sad attempt to feel like part of them while he's really an outsider. Sometimes it seems like he may have been there, and usually it seems like it was probably another Donnelly or sometimes Jenny who was really there.
How I Met Your Mother started off occasionally playing with this, but has used the device increasingly often as it progressed. Unusually, it is not because Future Ted is lying per se (at least, not often - there are some instances of outright lies), but because of ordinary memory lapses (having a character named Blah Blah because he can't recall her name), subjective interpretation of ordinary events (showing Robin's forty-something date as elderly), or sanitizing the story for his children (using "I'm getting too old for this stuff" instead of "shit".). The few times he tells us things that seem to defy reality (such as Lily and Marshall escaping their own party by jumping out the window, or having high school athletes and a Teen Wolf on a kindergarten basketball team), he Hand Waves it by saying that's all he heard about it. In short, if there is a way to exploit the potential of an unreliable narrator for comedic purposes, it's been done on How I Met Your Mother at some point.
Episode "The Rough Patch": Since they began happily dating, Barney and Robin have let themselves go a little; however, in Ted's mind, they look like absolute hell, and Barney in particular is now comically overweight. He even admits that he's unreliable on this point, but they stay that way for most of the episode anyway.
Episode "Zoo or False" includes two more examples. The question of whether or not Marshall was mugged by a monkey goes unanswered, and the last two minutes of the show, where the monkey carries a little doll woman to the top of Ted's scale model of the empire state building while paper airplanes are thrown at him are left similarly ambiguous.
Ted: Barney, enough with the lies. You can't just tack on a new ending because you're unsatisfied with how a story wraps up.
Barney: Oh really? Well, mark my words, Mosby, 'cause someday you'll be telling this story, and you'll see it my way.
Particularly great, since the setup for the awesome end has been laced throughout the episode—so if Future Ted is making this up he's likely made up a fair chunk of the episode.
This also happens to Ted when he goes to see a movie and finds out that the story is based on how Stella left him right before their wedding. It portrays him as a Jerkass and makes him the villain.
Speaking of Present!Ted's Jerkass behavior, Ted comes off as a Nice Guy, but continually done some pretty selfish things. Is he worse than he appears? On the other hand, Future!Ted tends to insult his past self fairly often. He seems to recognize his behavior as wrong and learned to grow up. Or has he?
Subverted in episode 5.5, "Duel Citizenship:" Future Ted says, "And then it happened... Marshall and Lily morphed into one big married blob." This is shown literally happening, indicating Ted's narration is being exaggerated for comic effect. Then Present Ted blinks and says, "Whoa...I gotta dial back on the Tantrum." This refers to a highly caffeinated beverage he'd been consuming, implying that he was hallucinating.
An ongoing joke in the series is that Ted doesn't want to admit to his kids that he and his friends occasionally smoked pot, so any time he refers to a joint, he calls it a "sandwich," and the characters are duly portrayed eating sandwiches (while their behavior makes it obvious they're stoned).
Telling Bashir how it was a fellow Obsidian Order agent named Elim who screwed up Garak's life. When Bashir finds Garak's mentor (and father) Enabran Tain, he asks about this. Tain just laughs and reveals that Elim is Garak's first name. In a way, Garak was saying that his predicament is his own fault.
Episode "The Wire": Garak, because as a former secret agent of the Cardassian Obsidian Order he liked obfuscating his own past and never told a truth if a lie would suffice.
Bashir: Out of all the stories you told me, which ones were true and which ones weren't?
The moral Garak draws from The Boy Who Cried Wolf? Never tell the same lie twice.
Garak: The truth is usually just an excuse for lack of imagination.
Dexter often mentions his lack of any emotions in his narration, though it becomes increasingly apparent that this is not true. He's not lying to the audience so much as he simply doesn't understand a lot of human nature.
In one segment of MADtv, Aries Spears tells a story as a photomontage of the events he's detailing accompanies. We start with Aries hanging out on the roof, where he goes to chill out in his downtime, and noting that this would be a great place to launch a glider. After this point, the wholesome and educational narrative he details begins to subtly (and, very very shortly, not so subtly) diverge from the things we're seeing, and ends with Aries high as a kite on glue fumes, under the impression that one of the other actors, aware of what has happened and concerned for Aries' safety, is some kind of demon out to kill him.
The Dharma orientation films of LOST are narrated by François Chau's variably named character. The Swan film is located "behind The Turn of the Screw" on the bookshelf, tipping the audience in advance that perhaps "Marvin Candle" is not to be trusted.
Hard to prove, but Kevin of The Wonder Years may fall under this. He is recalling events to him long past, and while the broad details are likely accurate, consider that the older brother and some of the pre-Women's Lib neighborhood girls get away with a lot of hitting. Also, when unfairness, especially parental, hits Kevin, it seems to focus on him exclusively, making you wonder if his older self is letting the filters of nostalgia and occasional bitterness influence his re-telling. The premiere episode has Kevin recalling that he was a 'pretty fair athlete' while showing a perfectly thrown football pass bounce off his chest.
Malcolm in the Middle plays with the more humorous variant. For one example, Malcolm says the house next-door never seemed to have a permanent resident and they never figured out why. Cue montage of the boys playing all sorts of pranks on the previous residents, then cut to Malcolm saying "I don't know - I think it might be haunted."
Episode "The Trial of a Time Lord", the Valeyard has tampered with the evidence in the Matrix, especially in Mindwarp, to make the Doctor's conviction certain.
In the more recent Doctor Who story, "The Unicorn And The Wasp", Agatha Christie questions the attendees at an outdoor party regarding a recent murder. As the suspects each give their story, we see the events that they describe, but as they really happened. Example, one young man claimed to be wandering alone, but in the flashback scene it's shown that he was flirting with another man. His father lies not only about what he was doing but also what he was reminiscing about at the time, leading to a flashback-within-a-flashback.
The episode "Love & Monsters" is framed as a story being told to the camera by Elton Pope. It's explicitly shown that his memory of how the band sounded, and how they actually sounded are rather different, which calls into question a lot of his interpretation of events.
BBC sitcom Coupling had numerous examples of unreliable narrators, notably pretty much anything said by either Jeff or Jane. But the greatest example of was in the third season episode "Remember This", where Patrick and Sally's individual recollections of how they met match in many, but not all details, to great comedic effect. In particular, the print of Munch's The Scream that the exceedingly drunk Sally remembers is revealed to be a mirror in Patrick's memories. When Jane turns up unexpectedly at Patrick's flat, the lads discuss the incident at the bar:
Steve (astonished): Why?
Patrick (equally astonished): That's the first thing I said to her, I said, "Why?"
(Cut to flashback)
Patrick (suave): Come in!
(Cut to bar)
Patrick: She just came in. I had no idea what to say!
In "The Unnatural" an alcoholic ex-cop tells Mulder how he encountered an alien posing as a famous Negro baseball player in 1947 Roswell; a story that even Mulder finds hard to believe. When Mulder tries fitting these facts into what he knows about the Government Conspiracy, the cop basically tells him to just shut up and enjoy the tale.
Used this trope very frequently, especially in the more comedic episodes, like "Jose Chung's 'From Outer Space'" and "Bad Blood." In "Jose Chung's 'From Outer Space'", one alien is named "Lord Kinbote" after Charles Kinbote, the unreliable narrator in Nabokov's "Pale Fire."
In Dollhouse, Bennett's memory of how her arm was crippled shows Caroline abandoning her to save herself. Caroline's own memory is later seen, and shows her trying to dislodge the rubble pinning Bennett, then explaining that as an employee Bennett can pretend she wasn't involved, and pinning her ID badge to her to make this more obvious before leaving. Which seems very thorough. The apparent implication is that Bennett's memory is incomplete. On the other hand, Caroline is the one whose memory is repeatedly and extensively tampered with, so there's room for multiple interpretations.
The Janitor from Scrubs is a pathological liar. He tells the most bizarre tales about his past and doesn't even keep track of what is true in them, if any at all. Or maybe he does but just wants to screw with you. The only thing we know about him for sure is that he had a bit part in The Fugitive.
(as Janitor finishes a story)
J.D.: Is any of that true?
Janitor: Somebody would have to read it back to me.
Played for laughs on Red Dwarf. In the episode "Blue", the crew travel through an artificial reality version of Rimmer's journal, in which he depicts himself as a brave, handsome leader and the other crew members as reliant on him for various things which, in reality, they're better at than Rimmer.
Occasionally used in The Middle. A scene will go surprisingly well, considering things rarely if ever go well for the characters. Frankie will then voice over "OK, that's not really what happened," and show the much worse thing that actually happened.
On NCIS Tony tends to embellish his stories. In a sad example he has been embellishing a story about a school prank for so long that he started to believe that his version of events was exactly what happened. When he starts feeling guilty and goes to apologize to the now grown up victim of the prank, the guy is baffled by Tony's apology. Tony was actually the victim of the cruel prank and the other guy was the bully. Tony realized that over the years he managed to flip the story in his head and made himself into the villain.
Alan Bennet's Talking Heads series of monologues is built on this trope. Each narrator tries to tell their story to their own advantage, but we can see through their facade to see the real story. For example, 'Her Big Chance' features Julie Walters as a woman who thinks she's a highly professional actress but we get enough hints to see that she is anything but (for example whenever she says a line, the director tells her it might be silent). She also appears to have no idea that she's acting in a soft-core porn movie for the German market.
The first episode of the fourth series of Misfits has a framing device of Rudy explaining the most recent strange occurrences at the community center to newcomers Finn and Jess. Each time they catch him in a lie, he backpedals and alters the story he's telling to avoid the relevant lie, admitting to cutting off Michael's hand with a hacksaw and conspiring with Seth to lock Curtis in the freezer and Lampshading his unreliability as a narrator (actually naming this trope outright in the process). Once he's run out of story to tell them, Rudy admits that he is only telling the story to stall while the drugs he has given them take effect, thus ending the framing device.
In the Stargate Universe episode "Twin Destinies", both Telford and Present Rush suspect the reliability of Future Rush's claims that he tried to save the rest of the crew after the accident. The later episode "Epilogue" reveals that at the very least he was lying about which crewmembers stayed behind with him.
The Teen Wolf episode Visionary is a series of flashbacks framed by Peter and Gerard depicting a tragedy from Derek's past and the events that eventually led to the formation of the alpha pack. What the flashbacks depict vary wildly in some places from what Peter and Gerard actually say happened and the trope is actually mentioned by name. Further complicating matters, Word of God states that while the audience knows more about what really happened than the characters, they should not assume they know the whole story.
The entirety of The Goldbergs is this, with every episode narrated as occurring in "nineteen-eighty something". One episode featured the wedding of Prince Charles and Princess Diana (which occurred in 1981), while also including the song Eternal Flame (from 1989).
The trope comes up repeatedly in True Detective. Hart accuses Cohle more than once of coming up with a narrative that might explain a crime and potentially bending the evidence to support the narrative rather than letting the evidence dictate the narrative. Both Hart and Cohle's present day accounts of what happened in 1995 omit and fabricate various details, and in a scene integral to the case in episode 5 their depositions directly contradict what plays out onscreen. In the same episode, detectives Gilbough and Papania call out Cohle on his unreliability in a big way.