Not too long after that, I found a locking nut. A few days after that, I found a washer.
All I had to do was figure out where they had come from. I poked around briefly and didn't see anything obvious, so I postponed the job until there was time to take a good look at the problem.
Tonight, I finally had time to look at it. And there was another Allen head bolt that was sitting in a screw hole with absolutely nothing holding it in. Gretchen pulled out the tray of assorted nuts, I found one that matched, and then we tipped the chair over on its side and reattached the back half of the reclining mechanism to the base of the chair.
The chair is much more stable now, which is good, because we have a SpaceTime Theater rehearsal tomorrow and working chairs are a good thing.