In the US, we're not ordinarily required to keep any sort of ID on our person. There are some exceptions, such as the mentioned airports, crossing federal borders (as in to Canada or Mexico), some federal facilities, maybe some state/local government facilities, and (state dependent I've learned) when operating a car. Otherwise, you're pretty much free to leave your home in nothing but shorts and maybe a shirt (public decency laws and all) and go almost anywhere without issue.

Real ID is irrelevant to this. The issue is that now they can demand that people prove their citizenship almost anywhere and anytime beyond the few places it was permitted before.