[schoen]

I've been told to carry a photocopy of your passport. If you're stopped by police and they insist on seeing the real thing for some reason (like to check your visa), you can tell them what hotel you're staying at and they can take you back there to get it.

While police might be annoyed by this, you may be unlikely to encounter a situation in which it's necessary at all, and it probably won't endanger you unless you're in a place where police habitually try to extort foreigners, in which case it could provide a new pretext for extortion.

The best advice probably varies by country, depending on local crime patterns and also on the attitudes of local authorities to visitors (or to visitors like you).