There isn’t one. But you have to admit that it would be interesting if there were.
There are all kinds of different twists on basic biology that would make for a different kind of society, if they only existed in humans. Some of these have been talked about a lot, served as the framework of various science fiction novels, etc. Most have not, because not many people know the biology – including most biologists.
Bees, wasps, termites, aphids, ambrosia beetles, naked mole rats, and a few species of parasitic shrimp are eusocial. That is, they cooperate in taking care of the young, overlapping generations live within a colony, and there are reproductive and non-reproductive castes. Usually, there is a single shared living space which is protected by some of the non-reproductive castes – the Place That Must be Defended.
Are humans eusocial? They do cooperate, sometimes – but in truth they’re not at all like the social insects. E. O. Wilson thought so, but then he’s kind of dim. No castes, no reproductive division of labor, while most cooperation is with non-relatives. I’ve certainly seen SF stories with this theme – Big Sword, Rogue Queen
Some pinheads have suggested that male homosexuals are a kind of caste, one that pays its genetic way by helping their sibs raise more kids. As far as I can tell, this idea originated with a student of E.O. Wilson. Since homosexual men do not actually support their sibs noticeably more than straight men, this is silly. If you’re going to invoke a behavior in support of some evolutionary hypothesis, the behavior has to actually exist. I shouldn’t have to say this. I could say that you expect mothers to work hard to take care of their offspring, because of kin selection, and because basic biology in mammals already requires that a female invest heavily in every offspring (not the case for males), etc, etc: but mother love ACTUALLY EXISTS.
One might as well talk about homosexual men as a natural soldier caste, born to die defending the nest. But they aren’t like that, don’t act like that, even counting the Sacred Band. And humans, especially in the old hunter-gatherer phase, didn’t usually have fortress-nests.
There are a lot of interesting twists in eusocial species that would make for even more interesting societies. There are harvester ants with two kinds of reproductives: females are AA or BB, males are A or B, but workers are all AB. It looks as if this is the residual of hybridization of two species. Apparently these ants have harnessed hybrid vigor, way before we invented mules.
There are cytoplasmically-transmitted thingies (microorganisms and rogue DNA) that distort sex ratios. Wolbachia infects many insects, and causes weird shit like male killing, parthenogenesis (curable by antibiotics), cytoplasmic incompatibility, and sometimes turning genetic males into functional females. Anything like this in humans? No, but it sure would be interesting if there were. Actually, cytoplasmic maternally-transmitted thingies might exist (they would probably be mutualistic), but if they do, they they don’t cause sex ratio distortion or parthenogenesis – we would have noticed.
Wood lemmings have two kinds of X chromosomes, X and X*. X* chromosomes ensure that the embryo is functionally female, whether the other chromosome is an X or a Y. So XX, X*X, and X*Y individuals are female, while only XY individuals are male – and are about 25% of the population. Sounds like a Beach Boys tune.
Some hybrid creatures, like our state lizard, are completely parthenogenetic. Again, that would make for a wildly different society. Virgin Planet, by Poul Anderson.
Many species have several different kinds of males (a few have different kinds of females as well). For example, a lizard species in California has three different kinds of males – aggressive orange-throated guys that successfully dominate blue males, sneaky yellow guys that get past orange males guarding a big territory, and blue mate-guarding males (that are also cooperative – possibly a green-beard gene) that successful guard females from sneaky yellows. The population frequencies oscillate: scissors, paper, rock.
Ruffs (birds) have three kind of males: territorial males, satellite males (literally ‘wingmen”, about 16% of males), and sneaker males ( ~1% ) that look very much like females. If we were like that, life would be complicated, but at least you’d know where you stand. I don’t think that anything similar exists in humans, though: right now there isn’t really any evidence that variation in human personality traits is adaptive, although it might be. Even if nonadaptive, it can be useful: you hire OCD guys for security and quality assurance, nymphomaniac receptionists, etc.
It’s not impossible that different Y-chromosomes have some effect on fitness in humans – they can influence behavior in mice – but again, no evidence for it right now. Never seen a story based on male morphs, but I haven’t read everything.
Green-beard-genes, originally suggested by Bill Hamilton, would modify appearance in a specific way and cause bearers to cooperate with others that bore the same allele. something like this occurs in fire ants: those with one version of a supergene have single-queen colonies that are hostile to each other, like Greek city-states, while those with the other version have multiple-queen colonies that extend indefinitely, rather like Los Angeles. In humans, a green-beard would probably rapidly lead to world conquest: perfect asabiya would be unstoppable. One more interesting application of CRISPR.
A species can have a weird genetic substructure, which may be hidden. Mice mate with mice that have dissimilar MHC alleles. Do people? Since it is a really fun idea, while the results of studies are mixed – probably not. Likely it’s all wishful thinking. It could be true though, or might be true in some populations.
White-throated sparrows fall into white and tan morphs with different behaviors. White males are highly promiscuous, while tan morph males are monogamous and contributed more to caring for offspring. The pattern is similar in females. In practice, practically all mating involves a male from one color morph and a female of the other color morph: effectively, this sparrow has four sexes. A similar pattern in humans would, I think, drive us all crazy.