23k sexes that's fascinating, so I looked it up:
They have two sex genes, named A and B, each of these have have an allele (alternate form) called alpha and beta. So combined there are 4 unique spots in the gene to look at.
A-alpha, A-beta, B-alpha, and B-beta.
Each of these can have variations, with A-beta having 32 and the rest 9.
32 x 9 x 9 x 9 = 23328 unique sexes
So describing your sex would look something like
A-alpha-1, A-beta-8, B-alpha-3, B-beta-5.
Honestly, just doing that would not only solve certain issues, it would remove some unnecessary complexity from English. In most (maybe all?) Turkic languages we don't have gendered pronouns, just one for "they" and nothing disastrous has happened yet.
Was curious as to the use of the prefix Schizo in the genus name and found this on a CDC website: 'Schizophyllum is derived from “Schíza” meaning split because of the appearance of radial, centrally split, gill like folds'