Assuming this is true, then the “eww, gross” factor wouldn’t come into play, presumably. So, playing along…
Fallout: major for her side of the family, and I’d essentially be disowned by my aunt, uncle, and other cousins. Very conservative Christians, and there’s already a lot of tension because she’s become a liberal free spirited hippie kind of woman. My older male cousin would probably physically confront me (and he’s into MMA), we’d both be subjected to a lot of angry fire-and-brimstone stuff, and we’d also get a lot of flak over our unrelated-to-the-situation liberal leanings. My side of the family no longer exists outside of myself so no issues on my end.
Not sure how friends would react. Probably a few raised eyebrows, but nothing more. My guy friends would be jealous, because several have asked me to hook them up with her for a date. My female friends would probably roll their eyes, but they do at any woman I date, so hey.
Legal marriage: Neither my state nor hers (though apparently if one of us was unable to have kids, it would be legal there). Both states also consider such marriages void, so we’d have to move elsewhere. Luckily, much of the east coast and California seems amenable to our situation… not bothering to look at the south or southwest, though.
Facebook: I think that listing ourselves as cousins and in a relationship/married would cause more issues than just friends finding out, honestly. Opens it up to distant family, acquaintances, and coworkers figuring things out and harassing us.
Changing names: Knowing her, she wouldn’t take my name, and after the expected vitriol, I wouldn’t be keen on taking hers. I guess we could assume new names or just keep what we have if she’s not too pissed at her family to jettison their name.
Overall: Not seeing too many upsides even if it was “true love.” She’d lose her family, and we’d both have to uproot, change careers and universities, and lose our social circles. It’d practically be like we’d have to assume new identities and start our lives over from scratch.