What can cause foggy/misty/gray water when water heater is turned off?
Could be that your water has some minerals in it. I noticed some cloudy water and a funny taste when I first moved into my house. I searched online for a water analysis company and for about $150 they sent me a kit to collect water samples and analyzed it for several minerals.

Turns out I have a VERY high calcium content. According to their findings there is no upper limit to the amount of calcium you can have in your water but it will make it taste funny and cloud the water/give it a funny taste. Depending on how old your hot water tank is you may have some sediment built up in the bottom.

If you haven't ever done it I'd recommend looking up a company that does water analysis, they'll normally send you a kit with a number of bottles and instructions on how to collect samples. It's a few hundred dollars but it's generally worth the peace of mind to know what you're potentially drinking/showering with. Here is an example of a water testing company. The price looks to be about $120 (plus shipping the test sample back to them).

Depending on what's in your water, you may want to flush your hot water tank as well. Here is a set of instructions on how to do it. I do it about once a year. It uses alot of water but it helps extend the life of your hot water tank and provides a more efficient heating of the water.