Should tourists carry passports all the time? (Moscow)

Dears, how are you?

I am going to Moscow for three weeks next month. I read at some websites that tourists should carry their passport, registration and plane tickets all the time because the police conducts random ID checks in the street and one might get held up and/or fined if he or she does not have it at that moment, is that true? I went to Russia for a whole month in Protected content do not recall even seeing this. I would not like to have such a problem, but I would not like to face the risk of losing my passport in some unfortunate incident either. Does someone have an experience or recommendation in this matter?

Many thanks,

Henrique