"Italy-based company" or "Italian-based company"

Solution 1:

In my view "Italian" itself suggests something of Italy or belonging to Italy. Therefore, I am not of the opinion that using "Italian-based firm" is acceptable.

For example:

Tata is an Indian company.

This sentence itself suggests the use of "Indian" as something referring to India.

Now the second case: Italy-based company. This would be correct.

For example:

Tata is a global company which is India-based.

Solution 2:

It seems a bit more straight forward if you swap a country for a city.
Melbourne based or Melbournian based...
Something that is Italian is from Italy, no matter where it is. Italy-based seems to imply that it is actually in Italy, not just having a history that is Italian.