OK - I've got the tile hung and grouted, and pretty much cleaned up the mess. Granted, it's not the world's greatest tile and grout job, but it's on a short wall behind the dishwasher and under a counter. This means that 1) it was really, really hard to do, all squatted down, and 2) nobody but you lucky blog readers will ever see it.
For reference, here is before:
It will be a little while before the dishwasher is sufficiently off-gassed enough to install. In the meantime, the towels are catching the water and the new tile is keeping the wall safe from mold.