I'm probably going to get heavily criticized for this, and don't necessarily recommend you follow my lead. My only "cooking" is to rehydrate instant oatmeal and freeze-dried entrees. I don't use any soap, because there isn't any really greasy residue. I just use a small amount of water to rinse the remains off the pot, and let it go at that. The pot gets sterilized the next time I boil water for cooking or tea. (Anyway, I choose to believe it is sterilized.)

I've never gotten sick doing this, but it might be relevant that I'm rarely out more than 2 or 3 days at a time.

I don't believe my way would work if you're doing "real" cooking, when you might have burned-on food or other residue to deal with.

The only time I've had digestive tract issues is when I went with a group of Scouts who insisted on washing everything in hot soapy water, followed by a quick rinse. I thought the cocoa tasted a bit soapy next morning, and about two hours later, hunkered down in the latrine, I was right - only time I've ever had an oral enema, but it was just as effective. That's when I quit using soap in the field.