You are starting with a false assumption. You state that sex is determined genetically. It is not, not even in humans. The default sex in humans is female. To get a male, the default structures have to be modified and transformed. A
You are starting with a false assumption. You state that sex is determined genetically. It is not, not even in humans. The default sex in humans is female. To get a male, the default structures have to be modified and transformed. A long process of links and dependencies have to occur before the fetus becomes completely male. In the absence of any of these steps, the process reverts back to the default female.
Human sexuality is determined hormonaly, after the zygote starts developing. What is determined genetically is what hormones will be produced in utero in the first few hours, but these are not absolute. The default sex of humans is female. If after those first few hours, the process of masculinizing is not started, you get a female. Any interruptions in the process, you get feminization. There are many other factors besides the and Y chromosome that come into play. The sex of humans can theoretically entirely be determined in the complete absence of the X and Y chromosome. In fact' many researchers posit that the Y chromosome in humans may eventually completely disappear. It is pretty useless. It will be replaced by a standardized fully functional X chromosome. There are mammalian species in which this has happened. The sex of the offspring is determined by other genetic factors besides the Y chromosome.
In order for a genetic system to work, there needs to be a default setting. Nature seems to prefer the offspring-producing 'egg' sex as the default. That makes sense as a fail-safe mechanism. If all else fails, what is left will still be able to produce offspring. Genetic donor sexes are really not needed. They just provide extra genetic material for the sake of diversity. And genetic material can be provided by means other than sperm.
However many'sexes' you have, somehow there has to be a process for one form to carry the offspring. Unless, of course, the 'egg' is simply deposited and abandoned, in the hopes that it is fertilized. The'sex act' would not be between two members of the species, but would be something done over the egg. In which case, any gender can deposit the egg, and any gender can add genetic material to fertilize it. Gender becomes something other than sexual determination.
But if you stick to an exclusively gamete-determined sexual identification, you need to determine which gamete produces the genetic contributor, which gamete produces the offspring carrier, and, well, what IS the third one? Nature has a habit of making the third one 'neutral' (that is, incapable of egg OR sperm production).
So however you decide to do it, you will not end up with three'reproducing sexual genders', you will end up with three different'functional genders' at a minimum, and maybe up to 6 combinations of X, Y, and Z. (Even more, if you specify that THREE gametes have to be donated. That is, viable offspring need a gamete from each of three different donors. So you need, for instance, XXY or XZY or ZZZ. This gives a possible 10 unique combinations. But now you have to decide which combinations are genetic donors, and which combinations are offspring carriers. There is perhaps no neutral, if it takes three gametes to produce a viable offspring). That is, it might result in functional genders of perhaps worker, leader, builder, reproducer and so forth. (Workers, for instance, might have more hands, leaders might have a bigger brain, and builders might have more legs). The only feasible purpose of having three form identifier gametes is to produce a different species functional forms for a different function.
Nature seems to prefer that, in such cases, one or more forms are infertile and incapable of reproducing. So your third 'gender' would probably be 'neutral'. You can definitely see the complications of this being left up to genetics, and not environmental, conditions. Species reproduction is now invested in only one of three identifications. If left only to the probability of mating, either that third is very prolific, or species numbers would decline.
So your answer is that you really need to re-think what your objective is in having three gametes.