A foreign area whose rulers are allowed to retain power in exchange for the protection of a more powerful nation is called

Its a Protectorate.

A foreign area whose rulers are allowed to retain power in exchange for the protection of a more powerful nation is called a protectorate.

Explanation:

A protectorate denotes a state which is formally independent but which is under the protection of another state. In practice, as a rule, the protectorate has lost influence in important areas such as foreign policy, military and raw materials.

Protectorates have been common along history, and were known far back in Greek and Roman eras. During World War I, Egypt became a protectorate under Britain's protection and control. Cuba was almost a protectorate under the United States until 1934.

Today, no states have formal status as a protectorate, but there are several quasi-protectorates, such as Bhutan, Cook Islands, Marshall Islands, Micronesia and Niue.