In fact, the US Department of State recommends that all US citizens have a photocopy of their passport with them, when traveling abroad.
Supposedly you get in shorter lines at the consulate if you have at least a photocopy of your passport available. You should be able to petition the consulate of your home country to issue you a new passport.I think it may be illegal, but I've heard from people who travel to places where it's sketchy for Americans to travel, that they report their passport as destroyed, get a second copy issued, then keep the first one.
The reasoning is that certain immigration departments see red flags if they see visa stamps from certain other countries, and you may get grief for having visited them. Cuba and the US used to be one, but cross-border rivalries are another. Knowing who hates who and presenting the right passport to receive the stamp would save you grief. Also if you encounter corruption/extortion you can schedule the next flight out and run.
With the electronic ones now I don't know how many places that still works.