I knew someone who was a ladyboy in (Muslim) Sumatra. It's very common there. Maybe more common than being a homosexual.
But he moved to Hindu Bali, and it apparently wasn't socially acceptable. So he changed to be a (slightly effeminate) man. He has a German 'bear' boyfriend/husband & adopted children.
In Thailand, too, the ladyboy culture is far more dominant than homosexual culture.
Camp men in Britain are much more likely to just live as gay men.
I'm not really sure the correlation between being a camp/effeminate gay man (as opposed to the hulking hairy type) and becoming transgender, and whether an Asian ladyboy would say that they identify as woman, man or third sex.
There seems to be a great desire now to insist that someone is transgender/the opposite sex. So if Joe the pensioner who harrasses women dresses up in women's clothes he must BE/have been a woman, even though he is now appearing in court as a man. Transvestism is no longer a thing, everyone must now have magically transformed into the opposite sex.