I think this is the best explanation about what was going on with the Mandarin.
"I don't think there is any real relationship between the Mandarin or any terrorist group.
Killian just happened to use a bunch of images that were already associated with terrorist groups and incorporated them into Mandarin to make him seem like he's connected with a bunch of terrorists."
I don't remember the world "Ten Rings" even mentioned once in IM3. I'm still sticking with my theory that the real Mandarin is out there somewhere.