A character is nominally bisexual but is almost exclusively involved with one sex during the run of the narrative.
Writing about sexual orientation, especially same sex attraction, brings a bucketful of implications and problems for a writer. Politically, the tug-of-war between prejudice and gay advocacy makes a character's sexuality hard to balance so that one group can feel represented without causing an uproar. There is also a balance between having the exoticism of an uncommon sexuality without alienating the target audience. This is where bisexuality comes in. While bisexuality has historically been caught in this crossfire with the result being an erasure of their existence, the end result has also been that bisexuality, when it does surface in fiction, becomes a tool to consolidate this dilemma.
While bisexuality in Real Life spans a huge spectrum that covers both ends on the Kinsey scale and everything in between, it is actually very rare to see a show that treats attraction to two genders as equally valid and important. Terms such as heteroromantic, homoromantic, biromantic, and aromantic exist because of the distinction between romantic and sexual orientation; characters that fall under this trope would thus be described as heteroromantic bisexuals or homoromantic bisexuals under such terminology, as opposed to biromantic bisexuals. However, there is one form of bisexuality in particular that holds a disproportionate presence in fiction and that is the kind of bisexuality that, when it comes down to it, lies closer to the orientation of the target audience. There are many variations to this, but key is to create some form of pecking order between the sexes, presumably in order to make the character more appealing to the audience depending on what gender and sexuality they are expected to have, while at the same time having the titillation, comedic material or diversity of "deviant" sexual behaviour. Of course, the prevalence of the trope brings some Unfortunate Implications for real life bisexuals; that in the end it's only one gender that matters to them and that their experiences with the other one are worthless.
A bisexual character who is written in this way usually treats the genders differently by one or more of these three aspects:
Time: Alice used to date or sleep with both sexes, but there is no indication that she does so now.
Actions on screen: Bob sleeps with both sexes, but the only relationships he forms are with women. Of course, heteroromantic bisexuals—people who are sexually attracted to both genders but only romantically attracted to the opposite gender—do exist in real life, as do homoromantic and aromantic bisexuals.
Tone/emotion: These two usually go together. If Alice considers her experiences with women to be wacky hijinks and her experiences with men to be lovestories, they are usually treated as such by the music, the other characters and the rest of the set.
Nene from Hyakko says she's bi, but she only shows interest in girls. Since it's a comedy Slice of Life aimed at men, this is probably intentional. In later chapters she finally hits on a guy, Torako's looks-like-a-Japanese-Delinquent friend - or is just being her normal Lovable Sex Maniac self, as her idea of hitting on him is to ask how big his penis is, since he's really tall.
Hellblazer: Constantine has been said to be bisexual, but due to Executive Meddling his experience with men has been limited to one very limited seduction.
X-Men villain Mystique. While she is often shown seducing men to get what she needs, and had a very dysfunctional relationship with Forge, her only meaningful long-term relationship was with Destiny, another woman.
In My Immortal, most of the guy characters are said to be bi (perhaps for Yaoi Fangirls), yet the only evidence actually supporting this is that Harry (er, "Vampire") once dated Draco in the offscreen backstory. The same thing can be said for Ebony herself.
Klavier Gavin in Dirty Sympathy. In the offscreen backstory he had slept with groupies often enough that he can use them as an excuse but both of his romantic relationships have been with men. It's noted In-Universe when Daryan figures out that Klavier is cheating on him when he hasn't slept with a woman for months.
Aral Vorkosigan in the Vorkosigan Saga in the time aspect. "Do you know your husband is bisexual? - He WAS bisexual. Now he's monogamous." Admittedly, given that his home is a very sexually conservative planet and that he is courting or married to Cordelia throughout the saga, there is little scope for him to believably display attraction to men. (Especially as most of the books are from the POV of his (straight) son.)
An implied psycho-analytical explanation is that Aral was so oriented around the military that he was more attracted to soldiers than to women. Cordelia, as a woman soldier, was the solution to his dilemma. (It helped that he actually liked and respected her, too.)
Also, Byerly Vorrutyer, despite his suggestive name, mannerisms and confirmed promiscuity with both sexes for professional purposes has only shown interest in women on the page.
The author of Vampire Chronicles had established that all of her vampires are bisexual, but none of the female vampires seems to have interest in other women.
Oberyn Martell of A Song of Ice and Fire, to some extent; he only has one lover throughout the course of the series, and it's entirely possible that it would be unsafe for him to bring a male lover to court. The only evidence we have of his other relationships are rumors and his bastard children, the latter of which really couldn't come from his relationships with men. A more straightforward expample is Daenarys, who has and enjoys sex with Doreah and Irri, but her only relationships are with men.
Alice was introduced as the token bisexual character, and it was indicated that she slept with men and women more or less equally. However, as time went on, her relationships with men were given increasingly little screentime, and were eventually phased out entirely. By the end of the series, Alice is in a stereotypically normative butch/femme relationship with Tasha, an former army officer, and identifies as a lesbian. One of Alice's last male/female relationships on the show? With a man who identified as a "male lesbian". The show seems to want there to be NoBisexuals.
Played with in the case of Jenny Schechter, who is established as bi from the first episode and whose first-season arc is mainly about her discovering it. Of course, she doesn't have many relationships with men on the show, but then again, it's implied that she only dated men before moving to West Hollywood and may be making up for lost time. Regardless, though, since she's a Depraved Bisexual she doesn't really do much to lift the show's attitude toward bis. And she identifies as a lesbian.
House: Thirteen's tendency to have sex with women becomes a source for fan service during her whole run, while her relationship is with a man. Later improved somewhat by having her settle down with a woman
Even after Todd from Scrubs was established to be bisexual, he was almost never seen hitting on or telling stories about his escapades with men.
Gossip Girl has Chuck Bass who not only is seen almost exclusively with women, but the TV series seems to reduce his connections to men to flirting and kissing.
Anna from Chuck falls under Bi the Way, but hasn't seen any on screen action with women.
Captain Jack Harkness of Doctor Who and Torchwood shows attraction to all sexes, genders and species. Throughout both series, he gets into several long-term romantic relationships with men and women. His Mayfly-December Romance marriages with 20th century Earth women, and their repercussions, are often discussed. He has on-screen romance with several men (including James Marsters!) and one very hot gay sex scene. In Doctor Who, the show went out of its way to make Jack's kissing scenes with Rose and the Doctor. During the first season of Torchwood, his relationship with Ianto was sexual whereas his past relationships with women were romantic, which had shades of this trope according to some fans, and later seasons balance it out a more.
The Doctor has been openly bisexual in the show starting with the Ninth Doctor (the Eighth in the Expanded Universe), but most of his love interests, even Girl of the Week ones, are women - Rose, Jabe, Martha, Madame de Pompadour, Astrid, Joan Redfern, Queen Elizabeth I, and River Song, with Sarah Jane finally made canon as a love of the Fourth Doctor retrospectively. His only major male love interests are Captain Jack, who had healthily mutual sexual tension with the Ninth Doctor but Ten heavily implies that he has lost interest, and the Master, with whom the Foe Yay is intentionally very extreme but a complete personality transplant would have to happen before anything could come of it. The only other times the Doctor has mentioned his interest in men it has been as throwaway lines played for humour - specifically, the Tenth Doctor once suggests that a male ally kiss him (he doesn't), the Eleventh Doctor kisses Rory just because he fancies him while still being Shipper on Deck for him and Amy, and Eleven also complains about accidentally making himself a robot boyfriend in a gag line.
River Song is confirmed bisexual by Word of God but she flirts with only two women (Liz 10 and a colleague in the library) in throwaway lines. She's more like Doctor-sexual.
The Clara incarnations are bisexual, although we only get one piece of evidence for this - how a big fuss is made about Oswin's first girlfriend, with Oswin using her name as a flirty nickname for Rory. However, Oswin herself says of that relationship that she was 'going through a phase'. No other Claras show interest in women at all.
In Skins, Tony Stonem and Cassie Ainsworth are seen having meaningless trysts with people of both sexes, but their only relationships are opposite-sex ones. Franky Fitzgerald also says she's "into people" when asked which gender she prefers, but only shows any kind of interest in guys on the show (Mini's one-sided crush on her in S5 notwithstanding).
Donna Freedman from Neighbours unfortunately turned out like this. Having taken the brave step of having a bisexual character and acknowledging this in a subtle coming out scene, the producers never explored her sexuality apart from a token peck on the lips with a female character
Jane from Coupling claims to be bisexual, though some claim it's just a bid for attention. Oliver calls her out on it in season 4, showing her pictures from one of his magazines, which she immediately cringes away from.
Possibly a case of Flanderization - there's no indication in season 1 that her bisexuality isn't genuine, and she seems just as eager as the boys to look at Susan's breasts in the first episode. By the middle of season 2 it's become something she only does for male attention.
A particularly problematic case in Glee with Santana. Throughout season one she is seen having sex with many male characters, notably both Puck and Finn, as well as being very heavily hinted to have been sleeping with Brittany. However, she later comes out of the closet as a lesbian, rather than, say, a homoromantic bisexual. This leaves Brittany as the only bisexual representation on the show, and even so the show tends to avoid the term "bisexual" in favor of phrases like "fluid" or "bicurious". Not to mention various comments by the creators and the show itself can be interpreted as outright biphobic.
Ellen from El Goonish Shive. Technically bi, due to the specifics of her creation (she's Elliot's Distaff Counterpart, after all), but is in a stable relationship with Nanase and eventually decides that she's basically just lesbian, after a traumatic "dream," thus checking off the time aspect. Of course, since Nanase is technically her first relationship, and since she is bi due to her specifically being created that way, it would be very unfair not to take her at her word.
T-Bob in Something Positive. He self-identifies as a bisexual but is only seen dating or displaying an attraction to men, the only indication of him being interested with women was a throwaway reference to him coming onto a woman offscreen. And she turned him down.
Homestuck has the trolls, for whom sexuality is not a concept - while not all trolls are bisexual, the lesbian troll is considered to have a girl-fetish more than an actual orientation, one male troll is hugely confused and disgusted when rejected by a male human on orientation grounds, and the social justice warrior troll has trouble distinguishing between male and female trolls at all. All this established, for a long time, the trolls disproportionately hooked up or experienced attraction in male/female pairings, with only a handful (Karkat, Vriska, Equius, Eridan) displaying attraction to both, and all of those had at least one prominent opposite sex love interest who dominates their development (Terezi, Tavros, Aradia and Feferi respectively) with their same sex attractions being exclusively gags that go nowhere. The pre-Scratch trolls had somewhat more indiscriminate romantic hookups (Rufioh having a relationship with both female Damara and male Horuss with no indication this was abnormal and Aranea and Porrim past relationship) and caused some interesting ships amongst the post-Scratch trolls (such as Gamzee/Terezi when Gamzee had been exclusively attracted to other male characters before), but even those catalysed around heterosexual pairings like Latula/Mituna, Kurloz/Meulin and Kankri/Latula.
1977: The Comic: Plan9's drummer, Robyn, is generally portrayed as heterosexual, but kisses her friend Troubles (aka Lorraine) and wonders. Later in the saga, she has a full-fledged fling with a lesbian police agent.
Yu-Gi-Oh! The Abridged Series does this occasionally. Joey Wheeler (who was straight in the original), shows attraction to other guys (Kaiba, especially), but he and Mai Valentine are clearly each others' love interests, fulfilling the tone/emotion aspect. The show also averts most Unfortunate Implications since Everyone Is Bi and Ho Yay is par the course. As an Abridged series LK is (mostly) restricted to the original footage, so there's little potential payoff anyhow. Not to mention that the creator himself is bi.