We've had a few members recently approach us saying, "I haven't joined your community yet because I don't want (fill-in-the-blank) to be able to see my activity there. Can I join anonymously?"
My advice to them so far has been to join as they normally would but use a nickname as a username (I have to manually approve everyone, so I still need to know who they are in order to approve them and give them the proper permission roles, as well as for our reporting purposes). Then, after they're approved, suggest they edit their profile privacy and hide their profile and email from the public.