Perhaps you have a male character who is visibly touch-feely towards another male character, while being similarly open in rejecting the advances of any female characters who come his way. When the other male characters ogle the resident Ms. Fanservice, this character hardly bats an eye, but he seems quite interested in what the Walking Shirtless Scene is doing. Other characters in the work may even question exactly which way this character swings, but never get an answer. The character's sexuality is simply never made clear within the work.
Yes, this character is Ambiguously Gay—they display much onscreen evidence of exclusive same-gender attraction, but nothing is ever explicitly confirmed one way or another. The ambiguity can range all the way from from blink-and-you'll-miss-it implications to all-but-confirmed.
This trope has taken slightly different forms throughout the past due to differences in censorship and what was permissible in the media. In many old films, characters were often given descriptions that may be perceived as code words for "gay". Such words may include "eccentric", "queer", "confirmed/lifelong bachelor" or other words that can just as easily mean their conventional definitions without subtext, hence why it's ambiguous.
In other cases where the censors may not have permitted any overt implications of homosexual attraction, creators would sometimes utilize stereotypical traits as a roundabout way of Getting Crap Past the Radar and implying a character's homosexuality. For example a male character might lisp, show and have an interest in fashion and musicals, have no apparent interest in the opposite sex, and live in a single-bedroom apartment with one of their "roommates". This particular permutation of Ambiguously Gay has largely become a Broken Trope due to contemporary media being more open to depiction of explicitly gay characters, many of whom don't necessarily conform to stereotypes.
As a result, modern instances of the trope are less likely to rely on stereotypes or code; common reasons for characters in newer works being considered Ambiguously Gay include a lack of interest in heterosexual relationships, overtly implied romantic or sexual interest in a character of the same gender, and In-Universe rumors about the character's sexuality.
Compare Hide Your Lesbians, wherein the relationship between two characters is ambiguously gay. Frequently is combined with Word of Gay, for cases when the character's homosexuality remains ambiguous within the work itself for various reasons but is deliberate. If the hints are overt enough, it can overlap with Transparent Closet, and some types of Have I Mentioned I Am Heterosexual Today?, in which the character's dubious sexuality is much more openly questioned. A similar trope is Have You Tried Not Being a Monster? for when the gay innuendo is used to define something supernatural, such as vampires or mutants.
A closely related trope is Ambiguously Bi, which is similar to this trope but with bisexuality as opposed to homosexuality, and often involves a character who often shows explicit attraction towards at least one gender in canon. Compare and contrast with Homoerotic Subtext, which is about same-gender characters who may or may not be attracted to each other, but nevertheless have gay moments which are often lampshaded by the work itself.
Not to be confused with Ho Yay, which is an Audience Reaction for fans interpreting any interactions between two characters of the same gender as homoerotic, whether or not it's deliberately implied within the text and regardless of what sexual orientation the characters may have in canon.
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- Budweiser ran one of their happy, feel-good ads where the first person that a male soldier calls to tell he's coming home is another guy who he also embraces first at his coming home party, even before his own parents.
- The Happy Chop could count, if it qualifies as ambiguous: "if you like small fruit... or even a big one!" and "can I say hard cheese on television?"
- Gil Thorp: Lini Verde, a flashy-dressing Glee fan who's also a clinch player for the basketball team. His sexual orientation hasn't been stated. The storyline involves a website that calls him unpleasant names, but the exact nature of those names is unrevealed (probably a good idea for all sorts of reasons). When one character suggests wearing pink to support him, she gets the response "This is about bullying, not ... pinkness!"
- It's Grim Up North London: Jez and Quin in the Private Eye comic strip. Maybe they're a stereotyped gay couple; maybe they're an equally stereotyped pair of upper-middle-class artistic types, who happen to share a flat.
- Lucy And Sophie Say Good Bye is a 1900s comic about two women who take a long time to part ways. Almost every strip has them kissing goodbye. Due to Values Dissonance it's unclear if this was intended to be Platonic Kissing or Queer People Are Funny.
- The WWE tag team Billy & Chuck were ambiguously Manly Gay, from being spotted backstage oiling each other up and doing bizarre stretches, to their shiny red trunks and matching personalized headbands, to their boy-band entrance theme. This became less ambiguous when they planned a gay wedding, on television — and then became outright subverted when they stopped said gay wedding to declare that the whole thing was a publicity stunt, and they weren't really gay, just Heterosexual Life-Partners.
- While The Brooklyn Brawler's onscreen persona has always been a tough guy, an Urban Legend amongst longtime wrestling fans is that Steve Lombardi is gay, rumored to be in a relationship with Pat Patterson.
- Long before Billy & Chuck, WCW had Lenny and Lodi, who were very ambiguous about their "very close relationship" - until Executive Meddling from AOL Time Warner killed the angle, ending it with the revelation that they were brothers.
- During a D-Generation X run, Triple H and Shawn Michaels seemed to pass off as convincing Heterosexual Life-Partners at the very least. They only even somewhat pass because most viewers don't remember the original DX, where there was very little heterosexuality to be had, between Shawn kissing Hunter on national television numerous times and Hunter's infamous "bi" quote:
Triple H: I'm "bi" a lot of things. Lingual ain't one of them.
- The Fabulous Ones. They were a Mr. Fanservice team marketed to adult women as Playgirl-style sex symbols (as opposed to The Rock 'n' Roll Express, who were a Tiger Beat team aimed toward teen girls), but some of their posed photos do seem to have a bit of Ho Yay in them.
- Goldust from the WWE, even though his gay overtones are primarily mind games he plays with his opponents. One notable example is his feud with Razor Ramon, where in one match he rubbed Ramon's chest and freaked him out.
- The New Generation.
- Layla El and Michelle are very close, and it was Layla who suggested they go to couple's therapy. And then there's Layla's blatant dry humping Kelly Kelly and Natalya on TV.
- Too Much had "Too Sexy" Brian Christopher and Scott "Too Hot" Taylor as tag team partners who were very concerned about each other's safety. Billy & Chuck's "wedding" angle was originally planned for Too Much, but was vetoed by Jerry Lawler (Brian Christopher's father) who was afraid his son's career would never recover. However, they found more success when the then-WWF repackaged them as the Pretty Fly for a White Guy team Too Cool, Grandmaster Sexay and Scotty 2 Hotty and put them with Rikishi, and it got over much more than Too Much had.
- Sasha Banks and Bayley sure are close, with Banks even telling Bayley "I love you, I always have, and I always will." They hug a lot.
- Round the Horne has Julian and Sandy: a couple of out-of-work actors, though they could turn their hand to anything. In one episode they were lawyers, leading to the glorious line "We've got a criminal practice that takes up most of our time." (At the time, bein gay was still technically illegal in the UK.)
- The stage version of An American in Paris has stylish song-and-dance man Henri Baurel, whose sexuality is questioned but never resolved in the text.
- In Bandstand, Jimmy ignores the female ensemble while his band mates ogle the women in New York, has a considerable wardrobe of plaid pants, and has some delicate flair in his mannerisms. The casting call and a cut scene reveal he is unambiguously gay, starts out with a slight crush on Donny, and his boyfriend was a fellow Navy man who died when their ship blew up.
- Be More Chill: Michael's relationship with Jeremy can be seen as this. They obviously care a lot for each other, are each other's only friends before the events of the musical, and Michael's song after Jeremy abandons him shows how dependent he is on him. And let's not forget how Jeremy called Michael his favorite person.
- Legally Blonde: The stage version has a scene where the cast tries to figure out whether such a character is actually gay... or just European. Turns out he's gay AND European!
- The Nerd: The script indicates that Axel is to be played very effete, and the ending suggests that he's trying to get everyone else out of the house so that he can share it with fellow theater devotee Kemp.
- The Play That Goes Wrong: Max really, really doesn't want to kiss Sandra. When Trevor ends up having to read Florence's lines, though, he's much more enthusiastic.
- Pokémon Live! turns the normally Camp Straight James more clearly into this. He's still as flamboyant as ever and jokes about "Don't Ask Don't Tell" (an old military regulation that banned openly LGBT people from service for years).
- Shrek: The Musical is full of this trope, from the sassy Donkey, to the prissy Farquaad, to the entire pride-anthem vibe of "Freak Flag".
Pinocchio: I'm wood. I'm good. Get used to it!
- The protagonist of Ken Laszlo's Italo Disco song "Hey Hey Guy". All we know is the guy on the other end of the phone is a man... and so might be the narrator of this male-sung song.