Yup, the problem comes back when I open the tables in the other order.
I think I finally figured out what it was though. After opening each table, I was deleting records. Turns out there was an old delete trigger on the parent table to mytable. In the original order of opening the tables, this parent table was opened first. When I deleted a record, it fired the delete trigger, which opened the child table mytable. Then, even though it closed the child table, when I went to open it, it somehow thought is was already opened (I think) and so opened it shared. So, when I changed the order, the child table got opened first, so when the parent's delete trigger fired, mytable was already opened exclusive. Anyway, I removed the delete trigger, and now everything works fine in the original order.