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This is not true. The GI specifications are very strict on where and how the protected product is produced. With Parmigiano Reggiano, for example, the cows producing the milk must be within the geographic area, and even their fodder must primarily come from the region.

If this happens, it's not within the rules, that's straight up fraud.

PDOs like Parmesan require all production and processing to happen within the registered area, and usually following some quite strict rules, depending on the name in question.

https://en.m.wikipedia.org/wiki/Protected_designation_of_ori... has some interesting details.

I find this hard to believe, given how protective Italian regions are over their fayre.

Do you have a source for this?