No Heterosexual Sex Allowed

To try to solve an overpopulation problem, governments (in fiction and real life) may limit the total number of children a person or couple can have; enforced with sterility, abortions, infanticide, criminal punishment, or social stigmas. Less frequently, a Dystopia may try to reduce (or eliminate, if other methods like cloning or bottle babies are in use) pregnancy by banning sex altogether. However, the absolute rarest version of this trope is where the Powers That Be encourage (or require) homosexuality in some or all of the populace in order to keep the population down.

This is different from a Cast Full of Gay, which is just a story where the majority of characters happen to be gay, or tropes such as Everyone Is Gay and Het Is Ew which are about fanfiction (where straight characters in original works are turned gay for the purpose of the story). Also differs from Situational Sexuality or from homosexuality being the result of a One-Gender Race. Of course, in some universes, Homosexual Reproduction is still a possibility.

No real life examples, please; sorry, we have to impose Wikipedia's "reliable sources" rule on this one.

Examples of No Heterosexual Sex Allowed include:

Literature

In Joe Haldeman's The Forever War, the Earth's government officially encouraged homosexuality ("homolife") to promote Population Control, which had caused worldwide food wars; eventually, they eliminated heterosexual reproduction altogether and employed cloning to perpetuate (and change) the species. With the time dilation the protagonist experiences he eventually ends up as one of the few heterosexuals and is regarded as aberrant.
In L. Ron Hubbard's Mission Earth dekalogy, the Rockecenter Corporation (which secretly controls the world) promotes homosexuality as "psychiatric birth control."
In Anthony Burgess' The Wanting Seed, heterosexuals are discriminated against while homosexuality ("It's sapiens to be homo!"), self-sterilization, and eventually even cannibalism are responses to overpopulation and limited resources.
In I Will Fear No Evil by Robert A. Heinlein, the government doesn't quite go so far as to forbid heterosexual sex, but they strongly encourage homosexuality for this reason.
In Charles Beaumont's "The Crooked Man," homosexuality became the norm but heterosexuality was tolerated for a while. Then a politician demonizes straights and they are forced to chemically change their orientation.
While it's not exactly encouraged directly, the wizards of Discworld (who are forbidden to reproduce as a means of preventing sourcery) apparently are willing to tolerate homosexuality among their fellows. At least, there's no outright hatred of the act mentioned in Unseen Academicals, which was the only Discworld book to even touch on the matter.