Is it possible to ferment my girlfriend's period juice, and turn it
into homemade red wine?

I'm curious.

It depends how sweet she is. Based on your posting, I'm going to guess
that it would make a fat, flabby wine with no taste to speak of.
Unrefined, and probably tart-ish. Bulk aging would be obvious.
Finally, I am going to guess that using a screw-cap is not in your
future.

Please find another hobby, perhaps one that requires absolutely no
interaction with the rest of the species.

Can't do it with yeast, not even with wild yeast that may come with the
juice. You'll need Devine Grace to pull it off.