Suzerainty
From Wikipedia:
[…] a relationship in which one state or other polity controls the foreign policy and relations of a tributary state, while allowing the tributary state to have internal autonomy. The dominant state is called the “suzerain”.
This seems to be a fairly historical term. There doesn’t seem to be many of these situations in existence now.
“Suzerain” is vaguely related to “sovereign.” It traces back to the Latin for “up” and “ward.”
Why I Looked It Up
In a book about the oil industry:
The transition was now complete from the days when the companies had unilaterally set the price, to the days when the exporters had at least obtained a veto, to the jointly negotiated prices, to this new assumption of sole suzerainty by the exporters.
In this case, oil exporters decided to raise the price of oil to the larger market, without consulting the oil companies. They effectively controlled how the oil companies related to the outside world.
Postscript
[Hussein bin Ali] suspected [the Germans] of plotting to depose him and end his Hashemite family’s suzerainty over the Hejaz.