Really? First time I hear about these birbs, that sounds really cool :D I'm off to Wikipedia to fact check brb baiii
Well after about an hour of reading I found plenty of examples of gender reversal examples and some cool behaviors but nothing regarding chromosomes seems easily accessible or even mentioned. Reproductive behavior seems to be one of the main criteria used to establish evolutionary relationships (aka the cladogram) but that's as far as the layman can find online. I didn't search in Scholar though.
https://pmc.ncbi.nlm.nih.gov/articles/PMC7198006/
Edit, lmao:
The wattled jacana (Jacana jacana) belongs to clade Scolopaci, and its karyotype has not been reported yet. It is an interesting species with a polyandrous mating system, in which a single female defends a harem of up to four males by aggressively excluding other females from their territory; males provide nearly all parental care (Osborne et al., 1977; Emlen and Wrege, 2004).
really all it should take is just the word "seahorse", most people have at some point heard about how the males carry the children there. Can't really dismiss that.