Stinky Washed Shoes

I can usually get a pair of running shoes and keep them smelling ok for a long time, even after profusely sweating from my feet. They only start to stink once they've been fully immersed in water. It either happens when I run through a puddle or after I wash them. It's so strange that washing shoes can make them smell worse. Too bad I can't just keep wearing dirty shoes everywhere all the time (although it's not uncommon for me to rock some dirty shoes).

Does anyone have tips on how to keep my running shoes from stinking after I wash them?