Unfortunately, we don't have any sources that could give us an answer to this question directly. Indeed, we don't have any literary sources from Carthage whatsoever, unless you count a handful of lengthy inscriptions in Greek that we presume were translated from Punic.

What we do know is that the Carthaginian "Empire," to the extent that it existed, was so decentralized that it might be better to describe it as a sphere of influence than an empire. Most persons living within the reach of Carthage's influence would probably have thought of themselves first and foremost as citizens (or residents, or slaves) of their respective cities, which cities had their own mostly-independent governments/identities and simply paid tribute in soldiers, goods, and/or gold to Carthage. This was probably especially true of the residents of other Phoenician city-states that paid homage to Carthage as the preeminent western Phoenician city and their sometime protector. I doubt that most of them would have described themselves as subjects of Carthage or residents of the "Carthaginian realm," as you suggest, though they would likely have acknowledged Carthage's primacy over their homelands if pressed. But that's mostly speculation.

It does seem likely, however, that by 200 B.C. or so large tracts of the North African interior were populated by a mixture of Libyan peasants/smallholders and Phoenician-Carthaginian smallholders/great landlords, and that most of these people would have understood themselves to be direct subjects of Carthage. Many Carthaginian citizens held lands in the interior, and it would be strange if their holdings were generally considered separate from Carthage in the way that, say, a Phoenician sister-city like Utica was.

Sources:

Dexter Hoyos's The Carthaginians Glenn Markoe's Phoenicians