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Many countries of the world have laws about carrying ID and the rights of police to check those IDs. Many of those laws also specify that a passport is the only acceptable foreign ID for non-residents. The frequency of which these checks are conducted varies from country to country, city to city, district to district, police officer to police officer.
It is a good idea when traveling overseas to have at least a copy of your passport with you at all times. I scanned the front page of my passport, printed out a credit card size, color copy and laminated it. It travels in my wallet all the time.
While it does not legally satisfy the carry your passport laws, it does satisfy most people who check ID from time to time, police, hotels, etc.