Giving that fact that Rome saw non-roman people as barbarians , how that applied to Carthage Empire? (that during the Punic War was bigger than Rome territories and had a professional navy) Did Rome gave them a "special" nomination/ treatment ? or they just called them Barbarians too? If possible ,i would like sources .Thats a big question in my head .
There were a couple of treaties between them. The main focus seems to be Carthage would recognise Rome as the main power in Italy and Rome would recognise the trade routes around the western side of Italy as Carthaginian. I think they were banned with trading with Sardinia as well.
When Pyrrhus attack Italy and Sicily there was little help from 1 to the other so these agreements seem to be hands off my stuff type deals, though a Carthaginian navy was patrolling near Rome to prevent a sudden sea attack at 1 Point I think.
They really did stumble into the 1st war out of no where really.