take passport with me when sightseeing?

According to Italian laws all non-Europeans must carry a passport.

There are many discussions here on this topic
A search will bring them up

You are required by law to have on your person
Carry it safely in your money belt
US driver’s license is not acceptable ID in Europe

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'm paranoid about Passports getting lost or stolen when with me.

If its in you money belt, that is highly improbable.

Your US driver license is not official "ID" in Italy

When we had to leave ID to rent an audio guide at Hadrian's Villa, the issuer made it clear (no English) that he could not take the responsibility of holding my real passport. But when I offered a photocopy of it, all I was carrying that day, he smiled broadly and visibly relaxed.

I bring my Drivers License with me just to leave for an ID when I am renting a bike or audioguide.

My passport is always in my money belt, worn under my clothing.

If you do decide to leave your passport at the hotel, take a photo copy of it and carry that with your driver's license.

This is a recurring discussion. Be it known to you, by going around without an original passport you would be infringing an Italian law. All Italians bring with them an ID card all the time (if it gets stolen it is a nuisance also for them to get a replacement); and a US driver's license is not an official ID in Italy.

But in most cases there will be no consequences, and if there are consequences they should not be serious, as at most you can be stopped and/or taken to a police station; if it becomes a nuisance and time is lost - well, you knew and it will be your fault. - But there are some activities where a passport is necessary, like buying a telephone SIM; few chances of success without one. I am afraid that also driving without a passport, the original driving license and IDP (that is the official translation of the driving license) may be risky.

In those cities, definitely use a money belt to avert pickpockets. Keep one credit card and a small amount of cash in your pocket/crossbody bag and the rest in your money belt under your clothes.

In addition to the above i agree. Keep it in the money belt. Our experience on nine RS Tour is that those who had to replace their passport was because they left them in the room safe or had their purse stolen after leaving it on the back of their chair.
Additionally my theory is if something happens and we cannot return to the hotel I want to have my passport and access to cash. We keep both in the money belt under our close. Be safe and enjoy.

Always a good idea to have a color copy of your passport.

It's the law in Italy to carry your passport with you. You may not need it for ID for general stuff, but the last time we were in Italy we saw multiple incidents where police were actively checking for ID for people getting off trains or near crowded spots. In that situation a driver's license may not work.

Your passport is your only legal ID in Italy.
It’s their law and you are only a visitor.
Do keep a photocopy as well, as that could be useful if you lost the passport.
Keep the pp in your underclothing moneybelt and it will be perfectly safe.
I’ve traveled to Italy many times and have been asked to show my passport a few times.

We took a day trip from Venice to Codroipo, about an hour by train.
Inspectors boarded train, asked for passports.
We did not have our passports.
Certain amount of finger waving, our faces looking like vino rosso.

Never again would we leave for a day trip out of Venice without our passports.

Thank-you all. Great advice here!
Will definitely keep passports in money belt when going out.