[sokoloff]

I hand wash our stainless pots and get (what I think is) the same. I think it’s a very thin film of something (not likely soap as I can get it without soap).

I use white vinegar to dissolve it about once a month (making me think it’s basic, possibly calcium carbonate). Pour in a cup or so of white vinegar and set the pan to heat. When it’s hot but before it boils, swish it around and then pour it into the next pan. Rinse the first pan in plain water while the vinegar sits in the second. When the first pan is rinsed, pour the vinegar into the third, etc.

Our stainless pans never see the inside of the dishwasher and still get this.

[exmadscientist]

Do you have hard water? That might do it.

Also, presumably you're fine with vinegar, but for anyone else who prefers less fragrant cleaners, citric acid (usually available as a powder) can almost always be substituted for vinegar with comparable to better results.