Here's how I fixed it. I'm sure this won't happen to anyone else!
When I was cleaning the globe I noticed that it was oddly top heavy, like something was stuck in the mysterious chamber that the litter goes into when it rotates. I took a garden hose, filled it up with water and sloshed it around. Out came a blob of litter the size of my fist. Obviously a lot of litter had accumulated in there so I tapped around on the top of the globe and loosened up a lot more litter that was stuck in there and flushed it all out. Now the globe is balanced like it was before. The globe rotates smoothly and I don't get the blinking lights when it's done.
Apparently the base monitors the load on the motor and if anything funny happens like the motor requiring more load than it should, it goes through the cycle the best it can and gives the blinking lights error when it's done.
That magnet thing is a good non-mechanical way of detecting its position. It has to figure out where the globe is since it could be in any position when you first plug it in.
I also have a lot more litter capacity now!