Monty Wild is incorrect in stating that "No such reproductive systems have been recorded on Earth." In fact, such reproductive systems have been recorded on Earth.
The most complex gender arrangements in real life involve fungi. Some fungi are tetrapolar (four gendered) and there are or have been tripolar fungi. It isn't clear that any tripolar fungi exist naturally today, but tripolar fungi can be and are bred as hybrids.
An evolutionary path to tripolar can involve going from bipolar to having the option to reproduce sexually or asexually, to having the ability to hybridize with another similar species with two somewhat different genders (often expressed A, a, B, b), to having to have all four genders to reproduce sexually, to having one parent that has two genders and one that self-reproduced and has one gender.
Another similar arrangement is called a patchwork virus:
Scientists found a virus that is made out of 4 to 5 separate
components - it infects mosquitos, and they have to catch at least
four of those components to get infected, the smallest, fifth
component is optional. For plants and fungi, similar viruses were
known before, but (at least according to the study) this is the first
example in animals studied in detail.
The number of patches in a patchwork virus can vary greatly. You could have a female analog that would correspond to the "host" and might or might not be sentient (Ringworld by Larry Niven, for example, has a species with both a non-sentient gender - perhaps historically a reproductive host of a parasitic species - and a sentient gender), and then the elements of the patchwork virus function as gametes.
One can imagine a scenario in which sex involving just two of three genders is purely recreational, while involving all three genders is reproductive. One functional benefit of this would be that reproductive sex would require coordination, cooperation and deliberation. Something similar exists in modern hyenas. In hyenas, both the male and female must have the equivalent of an erection at the same time which makes a rape pregnancy almost impossible in hyenas, unlike almost all other mammals. This gives females much more reproductive choice and they have utilized it to produce a thriving more intelligent and socially coordinated species than any other megafauna carnivore. A tripolar gender system could work in a similar way - insuring reproductive choice. Also, simultaneous tripolar reproduction is much sexier than sequential tripolar reproduction.
Another way to get multiple genders is to have gender determined by environmental conditions at the time of conception or gestation which is quite common in many species (including vertebrates). Usually the environmental conditions involve temperature or pH (i.e. acidity), but one could imagine humidity or all manner of other conditions (e.g. pheremones from existing community members reflecting current gender ratios) playing a role in determining gender.
There are also three parent humans in real life, some via intentional genetic manipulation of multiple parent sperm and eggs, and others alluded to by HilWithSmallFields involving a sperm gamate donor and an egg gamate donor whose fertilized egg is implanted in a surrogate mother who contributed the mitochondrial DNA as I understand the process (apologies if I am incorrect, but the three part split is sensible fictionally, even if it doesn't exist in real life). Also, most women who have children in humans are at least partially in some parts of their bodies, chimeras, incorporating some of their children's DNA (and indirectly, their partner's DNA) into themselves.
Of course, humans have various non-binary gender options. Most notably, gay men, lesbian women, bisexual individuals, transgender individuals (both homosexual and hetrosexual), and even a very rare group of individuals who cycle through the course of a day or a month from one gender to another (a bit like transgender but unstable). In addition there are people are basically neuter (castrates and people born sexually ambiguous). There is strong anecdotal evidence for femme and butch sub-identities within a lesbian sexual orientation that have a biological basis associated with levels of testosterone exposure in utero, and for the equivalent in gay men, at least some of the time.
The Netflix series Hemlock Grove has an edgy scene in which a werewolf, a vampire and a human woman have a menage a tois that illustrates the dynamics of what a trigendered sexual relationship could look like (compare Twilight which has another love triangle of that type). Notably, many versions of the tales allow vampires and werewolves to reproduce both sexually and asexually - an asexual reproduction option seems to be a good foothold into a trigendered system. The Vampire Academy series by Richelle Mead also features a complex gender system involving living vampires, dead vampires, humans and dhampirs. And, the Vampire Diaries series also has some serious gender/species complexities.
"Being Human" (a TV series with a British and American version of exactly the same script) doesn't have gender complexity to it, it does have multi-species households that could provide further insight into the social side of the relationships.
I have read an interesting science fiction short story in which individuals had sex via a "bridge" which could be transferred from one partner to another during sex changing the gender of the people involved. One could imagine a "bridge" that only activates a seed when a sufficient number of people have contributed to it.