Tile shower is leaking
Well, I didn't attract any other answer besides CoAstroGeek's comment, but we went ahead and started tearing out our shower anyway. We found multiple punctures in the pan liner (including some from mounting cement board to the inside of the curb), and general shoddy construction (big thanks to the Floor Elf for his helpful pages on building a shower).  I think I'm well justified in pulling everything out and rebuilding.  I guess I could try to patch the pan liner, but I'm never going to match that tile, and so I may as well rebuild everything from scratch, properly.