Hmmmm, does she give any reason for her not wanting people to know you are married or why she is inconsistent in this matter?
Have you ever actually told her how you feel about her not wanting people to know that you are a couple? doesn't have to be a setting of boundaries just a coversation. You can just talk about how it makes you feel without feeling you need to take a stand or make a boundary. No pressure for you then.
What was your wedding day like, a large affair or just a few close friends-just wondering IF she has always had this unusual need for privacy or if it is a recent thing and therefore could be guilt as maybe people know she was having an affair.