Most likely nothing but being a bit surprised. If I would discover she (I am a heterosexual guy, so only a female could do for me) was actually my sister (which has happened to people who never met their biological family before since birth) then there would be a risk of inbreeding... The more persons there are between me and my partner the smaller the inbreeding risks would become and eventually even neglectable. A more direct cousin because I never knew my (biological) uncle or aunt once cheated and therefore had a child nobody in the family ever knew about, could be a different story, and especially when I'd already have kids with her when I discover this, I may gather some information to see if my kids would be exposed to extra risks due to this. When it comes to 3rd or 4th, I think I'd care more about the question if I still love her for the things I loved her for before the discovery actually.
Laws banning sex with close relatives in order to prevent inbreeding and incest, vary already when it comes to direct cousins, and I believe when it comes to 3rd and 4th or even more do not carry a ban at all. And over the entire scope of humanity and its history we are ultimately all distantly related to one another, however do to the many generations and how far humans have expanded, it's eventually no longer possible to trace anymore. So strictly speaking whoever I have a sexual relationship with would always be a distant cousin somehow... But of course, I know you didn't mean to go that far. Like I said, if I still feel my relationship with her is good, I wouldn't care... And I would hope she would feel the same way.
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