After many failed attempts (detailed below) the following tactic worked. This damaged the pair that I used it on but as they were white running shoes and the material under the first layer was also white, I didn't mind.
- Get a high-percentage Isopropyl rubbing alcohol. I used 90% alcohol, 10% water. You can buy this at almost any grocery store or Walgreens/CVS.
- In a well-ventilated area or outside and FAR AWAY from open flame, douse shoes in the alcohol. They should not be soaked, but you should be damp. I used about 1/2 cup on each shoe. Might not need this much.
- Wrap in two plastic bags.
- Wait a few days.
- Open the bags (which will stick a little to the shoes), set the shoes -- as open as possible -- in direct sunlight for another day or two.
- Soak in soap for 3+ days, leave out in sun for several days. Smell came back almost immediately.
- Soak in diluted chlorine-bleach for 2 days, leave out in sun for several days. Smell came back almost immediately.
- Anti-mildew spray I bought at Home Depot: Concrobium Mold Control. Worked for a few days, then the smell just came back.
I did not try this, but might soon. Feedback on this process is welcome.