The relationship between Carthage and the North African cities varied wildly. Much like the Romans in Italy, the Carthaginians maintained control mostly through economic power and the threat of military force. These cities were bound to Carthage through marriage ties and civil alliances, often being required to supply tribute to Carthage in exchange for Carthaginian protection.
Certainly, the cities were not always happy about the arrangement, particularly during the First Punic War, when heavy taxes were raised on the towns in order to pay for the armies that the Carthaginians raised to fight the Romans in Sicily. They also supplied much of Carthage's military forces in the form of levies.
The heavy indemnities imposed by the Romans after the war, as well as Carthage's many military defeats, contributed to the so-called "Mercenary War", when the armies of Carthage's vassal states and various mercenaries hired to fight the Romans rebelled after not being paid. Many of Carthage's allied cities joined the rebellion, resulting in essentially a civil war.
The situation varied city by city. Some were colonies and highly loyal to Carthage itself. Others were willing subjects who desired the stability and prosperity that Carthaginian rule could offer. Still others were subjugated tribes or tributaries who were only kept in line through the threat of military force and exclusion from the Carthaginian sphere of influence, which would leave them vulnerable to attack.