Confirmed - It is the water. To prove it, I took the same water that I use for my plants and boiled it out in a pot. Look at the bottom of the pot now:
The odd thing is that this residue tastes salty, where the film on the clay pots had almost no taste, except a drop of sweetness. Maybe the clay "absorbs" the saltiness?
Likely Na is washed out since it more soluble leaving Ca, Mg behind as a carbinate and/or sulfate depending on the water. The sweatness, well who knows what other compounds are left in the residual. You're not dead so it;s not to bad.