I looked on Google.com Advanced Search for "problem" "anal bead" but found so many articles that I couldn't go through them all. Perhaps you'll have better luck. Several different designs of these bead strings were pictured, and some of them looked dangerous if they should break.

You could eat a lot of soft foods like apple sauce and other foods that you know can produce soft bowel movements, and wait for results. A bulk-producing laxative might also work. In my opinion, a Fleet's enema might be too drastic.

If there's any possibility that part of the broken bead string might be poking or scratching the delicate rectal tissues, you really should give up and go to the doctor. Tell the receptionist that it's a very personal issue, and when you see the doctor, ask him not to keep any detailed information in your records. He could call it "irritation" or "constipation" or some such thing. If you explain your special privacy concerns, I'm sure he'll understand.