You did realize millions of people camp without riding a bike? We don't have showers then either.

First, I wear wool jerseys. They don't stink when wet, just smell like wet fabric, and believe it or not, for at least 3 days they smell like they just came out of the dryer when they've dried. I've not worn one more than 3 days straight -- they may not stink ever, but I've always managed to wash in a sink somewhere. But honestly, at home I often don't wash the jersey when I get back from a ride. I hang it up on the back porch to dry and let my wife (the one with the "nose") check it before I hang it back up. She can smell yogurt that's a day past the sell-by date from the grocery store parking lot so I know I'm covered.

In fact, I commuted in one today and it is hanging downstairs in my private shower. When I get ready to ride home it will be dry and indistinguishable from when I put it on this morning (it was actually clean then!).

I've done credit card tours before with a standard lycra jersey. Of course it smells like a hobo by lunch time. I just wash them out in a sink every night. I only bring one, the one I'm wearing. When camping, if I feel the need to wash something it's in the creek, no soap. That seems to work just fine. I wash the bibs only if I'm washing my own body, which certainly doesn't happen every day when camping, bicycle or walking.

I guess the answer to your question is: I don't worry much about staying clean.