Showing Passport

Yes. NORMAL.

Not only normal in Italy but in my experience everywhere in Europe.

We showed our passports 3 times during our visit to the Alhambra a few weeks ago, without leaving and coming back in.

....and in Italy, it is not unheard of for authorities for ask for your passport when out and about. While rare, there have been more reports on various travel boards of people being asked, and in some cases fined for not being able to produce their passport. Carrying your passport on your person is required in Italy (or rather having official ID), though many do not and get away with it.

We were just told by our RS tour guide here in Italy that it was ok to carry a picture of our passport on our cellphone if we wanted to leave the actual passport in our hotel room.

Edit: Yes, seriously to responses below. We are on a tour, we are not driving a car, we are in a group and if on our own for a while, we are not far from our hotel. And after twenty five years as a tour guide in Italy, that lives in Italy, I would hope she knows her stuff by now versus someone commenting that lives in the US.

Seriously?
Will that tour guide pay your fines if you happen to get asked for your passport while out and about and the policeman wants to see the real thing?

Recent discussion
https://community.ricksteves.com/travel-forum/italy/carrying-our-passports-in-italy

The Italian government is considering asking for a long term permit to stay to buy a SIM card, so it is possible the passport will not be enough in the future.

Next time you're stopped by a cop for speeding, just show them a photo of your drivers license and see what they say.

Seriously, a passport is the only legal ID for foreigners in many countries. We use state-issued drivers licenses in the US but they're not legal everywhere. We dont have national ID cards as they do in many countries.

Sometimes you get away with any photo ID for non-important activities, but that doesn't mean it's legal. Sometimes clerks are too busy to care, same as in the US.

I'm headed to Rome in two days, and yesterday I received an email from the tour provider for my Colosseum tour. Here's what it said:

The Colosseum requires all visitors (including children) to show a valid Passport / National European Union ID that matches the name on their ticket. Please ensure that you have your documents with you when you attend your tour, or the Colosseum will refuse your entry. No refunds will be given if your entry is refused.

I personally won't take a chance that they'll accept a photo on my phone or another form of ID. I'll just carry my passport as safely as I can.

there are 4 places where you are always asked to show a passport: airline check in and immigration, sim cards and phones due to control of phone usage, hotel check ins (never leave the passport with them, I carry copies if they need to complete their paperwork later. We once found our passports casually tossed on a counter in an unsecure hotel lobby with noone at the desk) and banking. You can be asked to show one during train travel -- this has happened to us three times in 60 years of European travel. Twice we were crossing borders although Shengen doesn't have routine border checks; the other time I needed to show my grandchild's passport to establish that the ticket was hers -- I think because it was an adult ticket.

You can decide if the risk of carrying it with you at all times is outweighed by the risk of not having it if needed. I have observed the disaster a lost passport can create so I don't carry mine out and about and haven't for over 60 years of European travel. I figure if it is a big deal I am more trouble than the need to fetch it from the apartment. I always have ID. If I am traveling out of the town where I am staying, then I carry it in a money belt under my clothes.

A fifth case: with nominative tickets for timed entry to sites such as the Colosseum or Vatican museums, they require to see that the name on the ticket matches that on your passport.

There have been many colloquies on this forum regarding the legal need to carry one's passport on one's person at all times in Italy. To the best of my knowledge, none of the participants has been an actual attorney versed in relevant Italian law. Until such a person appears, I believe what we have appearing on screen is more or less underinformed opinions. Mine is one of those.