Nothing to do with French seems a bit strong. It's related. From Brittanica:
> lingua franca, (Italian: “Frankish language”) language used as a means of communication between populations speaking vernaculars that are not mutually intelligible. The term was first used during the Middle Ages to describe a French- and Italian-based jargon, or pidgin, that was developed by Crusaders and traders in the eastern Mediterranean and characterized by the invariant forms of its nouns, verbs, and adjectives. These changes have been interpreted as simplifications of the Romance languages.
Heh, TIL, thanks. Obliquely, I was in Venice some years ago; sitting on the steps of a church I set to rolling a cigarette. A couple of small boys stopped and stared at this activity, one pointed and said "Il fabricato fumer!", I knew exactly what he was saying (although I have no Italian). So Venetian it is.