For some reason, when I hear "real woman", I think of all the nonsense about whether somebody (male) is a "real man." I know "real man" is mostly used against cis men, and I've heard "real woman" used against cis women. (It's never "for" anyone, except maybe oppressors.) And it's about gender policing. I suspect the same is true when it's used against trans men/women -- it's about keeping you in your assigned box, your assigned presentation, your assigned behavior.
Because once you open the door to people crossing over to a different box, you open the door to all kinds of uncomfortable things. It might make it obvious how poorly your "[all] men are X, [all] women are Y"-based rules actually fit the humans you deal with, and you'd have to base how you deal with each one on who and what they are as unique individuals. What??? Deal with people as they actually are, and not according to what category you've assigned them to? What a concept!
But you already knew that.