Lots of good ideas here.

We start by not getting very many dishes dirty. Lunches never call for dished on our trips. Breakfasts are instant oatmeal and cocoa...that means that the water gets boiled in our pot, poured into the oatmeal in the bowl, cocoa in the cup. WE rinse those out with some of the warm water from the pot (and usually drink that rinse water---hydration is good!) Then the bowl and cup get a very quick rinse with a tiny bit of Dawn, another rinse with water, and off we go.

Dinners are miso soup in that same cup, and then freeze dried dinners in their pouches. The pouches get packed out (as does all of our trash, from used TP to tea bags, etc) and the cups get washed as before.

As noted by others, always wash dishes far from any water source, and when it is time to throw the last few drips of water away, fling them so that they spread out on the ground.

Washing clothes? We rinse our clothes in water...but we don't wash them in soap.