This also happens in New York— one can buy an apartment with a rent-stabilized tenant in a building which has been converted to a coop. The tenant has a right to lease renewals until they die (and any family living with them at time of death has a right to lease renewals as well).

Here is one: http://streeteasy.com/building/243-west-98-street-new_york/7...

I'd be suspicious of a building that uses Comic Sans on its awning.

What is the Italian name for this type of contract?

nuda proprietà

That's what I supposed (I am Italian myself) but it looks different from the French thing. From what I understand with "nuda proprietà" you buy the apartment but cannot use it until the person in residence dies or voluntarily leaves. But you the owner will not pay anything to the "tenant". So there is no extra spending even if the person using the place lives much longer than expected.

Well nuda proprietà is actually a bit more general a concept: it just means that one person has ownership and the other has all the other benefits. Remember when Berlusconi touted that as the solution to his conflict of interest?

Don't we just call this a reverse mortgage in the US?

from my understanding, reverse mortgage is a mortgage where the owners slowly encumber their property with debt. If the owners die (or decide to sell), the property is still the owners, but it has the mortgage debt on it so they have to pay that off as part of the process.

these days, it is better to just move into a smaller flat and rent out the larger property at market rates.