It manifested the way that the Aurora instances would use up their available (meagre) memory, then start thrashing, taking everything down. Apparently the instances did not have access to any temporary local storage. There was no way to fix that, and it took some time to understand. After having read all the little material I could find on Aurora, my personal conclusion is that Aurora is perhaps best thought of as a big hack. I think it's likely there are more gotchas like that.
We moved the database back to a simple VM on SSD, and Postgres handled everything just fine.