Must a non-resident foreigner always carry their passport with them when in China?

When I am on a short visit to China, I stay with family and have to visit the nearest police station to register my presence. At the police station I provide my passport and write the address I'm staying at. They then give me a document (I don't know what it's called) with some information about me and my residence on it.

Now, I have read that foreigners need to carry their passport at all times with them when in China. But that's risky when I could leave it in a safe at my residence. Is the form that I get from the police station an alternative form of identification?

Also, 'free' museums and art galleries often require identification to enter; will I always need my passport or can I use the alternative identification given to me by the police?

If I don't have my passport with me and need it, what would be the outcome if I was stopped by someone in authority? My mother-in-law is paranoid that it will be stolen or get lost and is reluctant to give it back to me if my wife passes it to her.

Please note, I'm not talking about needing to travel anywhere outside of the city I'm staying in; I know the passport is necessary for flights or rail travel, registering in hotels, etc.