Everytime I depart Italy with a ticket on AZ and a separate flight on another Skyteam partner I am told they cannot check my bags all the way through to my final destination because I have two tickets.

What is the definitive rule here? Because leaving from the US I never have this problem.

Are the tix on the same PNR or were they booked separately?

Although not specific to AZ, my experience is that they don't have to, but they can tag a bag all the way through on split PNRs. My guess is that you're just experiencing the quality AZ customer service at its finest. A single PNR certainly makes it more likely to get things tagged all the way.

no, they were different PNRs... what I don't get is why they can simply add a flight to a PNR in the US and they can't do this in Europe, insisting that they cannot accept two different tickets.

Is there a firm rule about this?

If you are traveling on 2 separate tickets on 2 different airlines, you can't have your bags transfered on to the 2nd plane because you have not been issued a boarding pass yet for the second flight of that different airline. The first airline can't issue that boarding pass. If it could, terrorist could checkin with the first airline and have a suitcase placed onto a second flight of another airline without boarding the flight without having even checked into the 2nd airline.

One has to have been issued a boarding pass before your bags can be put on a flight. When one flies seamless on one airline or a partnered airlines, the airlines can access all your flight reservations and can therefore issue boarding passes for all connecting flights.

Each airline uses their own propriatory computerized reservation system that can only access their own reservations and cannot pull up and act on the reservations of another airline, except partner carriers and only under certain circumstance.

In the current security situation, when connecting between different carriers on separate tickets, one has to collect one's checked bags off the carousel and recheck them in with the second airline, even with partnered airlines.

If one's flying on the same airline but on 2 separate PNRs, tickets. The airline can see both PNRs and link them and your bags can tagged all the way to your final destination.