We make everything so complicated in our "labor-saving" civilization! I don't wash my own clothes or bedding! I send them to my launderer! The creek is my launderer! I place my clothes, blankets, and sleeping bag in the creek, weighed down by large rocks, leaving them in the water overnight. In the morning I wring them out and spread them on the rocks or on sage-brush bushes or on juniper trees. No soap, no scented dryer sheets, no polluting toxins, no fuss. And my clothes smell earthly-heavenly.
Silt sometimes collects on the laundry in the creek. No problem: I just swish the laundry around in the water and the silt goes away. The silt itself is a deodorizing conditioner.
This automatic laundromat doesn't get whites very white and doesn't remove some stains. But that's no problem since I wear mostly dark and earth-tone clothes, so such stains don't show. If you look around, white creatures aren't adaptable to the desert. Ravens and coyotes and bobcats and golden eagles are. But, if I may boast, my clothes are cleaner than anybody else's I know of (with the exception of my friend, mentioned below), because they have no synthetic chemical residues with off-gassing ghastly odors.
When I'm on the road, I wash clothes in bathroom sinks, park spigots, or in canals or streams. I try to wash socks and my feet every day, so I usually have a clean pair drying and a clean pair on my feet. I do often use soap to wash clothes in civilization.
I do have a hermit friend, who lives outdoors in New Mexico, who wears only white linen clothes (which he makes himself), and he lives by a very muddy creek. He refuses to use any soap. He washes his clothes in the clay by the creek and, somehow, they turn out white! He is an impeccably-dressed clean freak, in fact, and will not be caught with stains or stink!