Pax Night and day you can find me Flogging the Simian
Do what I do - photocopy (xerox) your passport and then put the real passport somewhere super safe at home or elsewhere. Then carry around your photocopy. Trust me, if the police stop you the photocopy will suffice and if there is some super emergency or something, they will let you get the original document.
Losing your passport and getting it replaced is a MAJOR hassle. Make several photocopies and stash them in different places or leave them with friends. It'll REALLY help getting it replaced if you lose it.
Also, usually you can get some kind of local ID with the Interior Ministry. Which country are you in? I use a Romanian ID here for banking and staying at hotels etc and the only time I use my passport is to actually go in and out of the country.