For instance: you register using your real name/"authentic identity", but you select a restrictive privacy option that says that your profile won't show up when people search for your name on Facebook; nobody can friend-request you; and all your posts and other info is visible to friends only.
Voila, you now have a private profile, but the people whom you are friends with on Facebook don't have to create a user script that replaces your pseudonym with the name they actually know you by.