I converted my tank to TSP a couple of weeks ago and I have done several parts in it so far. I have at least 8 parts that have been out of the tank for about a week. They have NOT been primed and have NOT flash rusted. We are on city water, but I installed a whole house filter several months ago. When I was using Washing Soda, my parts flash rusted within 2-3 hours. While unpainted parts will rust over time, the amount of time depends on local conditions. I can think of 2 things that might cause rust to appear as quickly as you are seeing.
1. As you and others suspect, something in your water. Try rinsing the part after you take it out of the tank with bottled or distilled water and see if it still flash rusts.
2. Could something have contaminated your solution? When I converted my tank I pressure washed it to get it nice and clean.