No Audio, Samsung Galaxy S4
Providing what worked for me, in case someone else has the same problem:

A helpful suggestion over at xda-developers.com by developer bilgerryan indicated I should Odin a full stock ROM, in order to replace whatever corrupted data was causing the problem.  Long story short, in the process of looking for progressively "fuller" stock ROMs after several failures, I ran across this thread, which provided a tar which "contains the NON-HLOS.bin modem.bin sbl2.mbn tz.mbn aboot.mbn rpm.mbn sbl3.mbn".

Flashing that tar fixed the problem completely, and since it doesn't touch the system or data partitions, it didn't require backing up or restoring anything.  (Although you should always back up your data before flashing anything, just to be safe.)  After that, I re-updated the NON-HLOS.bin and modem.bin to the further updated versions I had installed when sound wasn't working, and it continued to work.  So the problem was in one of: tz.mbn aboot.mbn rpm.mbn sbl2.mbn sbl3.mbn

Huzzah for having a working phone again.