"What is that odd thread?" I asked. I picked at it a bit and found this.
It's a loop from a join that got caught and didn't get pulled all the way up. I didn't notice it right away because it was lying so close to the previous ring. I briefly considered cutting the threads. At this point I had made both rings and chains with both shuttles after the goof, so cutting both would have been the easiest way to get back to that spot. However, that would have given me four ends to deal with, so that would have been just as much trouble as unpicking all the way back, and probably wouldn't have looked as good. Since I had only done part of one repeat after the goofy join, I decided that unpicking was worth it.
When untatting, I find it most efficient to make the threads much shorter than I do for forming the stitches. I also find that untatting causes a bit of twist to form in the threads, so I let the shuttles dangle frequently. Once I am done, I wind the threads completely back onto the shuttles and let them relax for a while. Letting the threads just sit on the shuttles for a couple of hours takes out the kinks so they are easier to work with when I do start retatting; plus, after untatting a long way I'm usually so irritated that I need to take a break anyway.
The ring with the goofy join happened to be right where I started this morning, so I literally had to take out every single stitch that I tatted today. Grrrr. But it is fixed now, and I am ready to retat and move on.