Other forms: spheres of influence
A region or nation that another country informally but powerfully controls is its sphere of influence. After World War II, the Soviet Union's sphere of influence included countries in Eastern Europe.
The phrase sphere of influence was first used in the 1880s, in reference to the political control exerted by England and Germany (and their rivalry) over various African countries. Today you can also use it in a non-political way, to simply mean an area over which some person, group, or business exerts power or authority. Sphere is from a Greek root meaning "globe or Earth."