Kept in secret maybe? I know that it wouldn't affect the inquisitor's reputation as much as Josie's (since she deals with the nobility so often,) but the inquisitor is the one who closed the breach and killed Corypheus, so they are already legend. Some Orlesian or Antivan gossip won't affect that. Her ties to the inquisition could still be an anchor for her rep., but I think overall it wouldn't be the smartest idea for Josie to hook up with somebody that isn't a noble. On the other hand, if she is in love with the inquisitor, it could add to her power and reputation regardless of origin because they are seen as the legendary hero who killed a tevinter magister and mended the sky.
Good point, but the way I see it, after the defeat of Corypheus, many will be quick to forget all about the Inquisition's heroics - "but what have you done for us lately" and all that. I imagine that in a couple of years, the elven Herald will be a "knife-eared savage" and the Qunari one an "oxman brute" again for the general population. In this scenario, it wouldn't do Josies reputation any good. Besides, aren't the nobles all about heirs? This would make her relationship with female or non-human Inquisitors even more complicated. Of course, marrying a respectable noble lord and keeping the Quizzy as a lover would be an option, but I don't see Josie ever being okay with that. I agree that this should have been addressed in the game.