More than anything else, I was wondering what this would look like.
My $0.02? Notoriety
Much like how Obligation is wholly negative, Notoriety would be an abstract numerical representation of how public and visible the character is. It could represent rumors, eye-witness accounts, and an ISB record. Maybe it's entirely unjustified, and you're playing that EotE character who's never seen a Jedi, honest, but it doesnt' matter because you got pointed out once, or was just won too much in that card game, or something.
Similarly, it would bring out Bounty Hunters and Inquisitors trying to hunt you down, or whether or not you're recognized in a populated cantina.
My only issue with this is that it'd be too simliar to Obligation, only you can't pay it off. Plus you can't hang character hooks on it beyond "Keep your head down. Don't cause trouble. Don't rock the boat."