How to recover "hidden" games after rebuilding my PS Vita's database?

I rebuilt my PS Vita's database from Safe Mode and most of my games were missing when it rebooted, even though they were still taking up system data according to the system's content manager. I had about 40 GB of games, but when I look at storage usage through the content manager, it shows 40 GB as being used by the system, and almost no space being used by PS Vita Applications. How can I recover my games without losing my saves?

This happened to me recently - I rebuilt my database to try to recover a corrupted save (didn't work), which only took about 20 seconds for some reason, and after restarting, only about 10 of the >100 installed games on my 64GB memory card were available, even though the card was still full. The other games seemed to have disappeared, and restarting the system again didn't help.

What ended up working was rebuilding the database again, which "freed" one more game that hadn't been visible before, and then rebuilding it a third time. This time, the rebuild took about 10 minutes, and then all of my games were visible and playable again. Hopefully this helps someone else who runs into the same problem.