Final answer:
The tibialis posterior muscle from the deep posterior compartment of the leg is responsible for inversion and plantarflexion of the foot, but for pronation of the forearm, muscles such as pronator teres and pronator quadratus are involved, though not part of the deep posterior compartment of the forearm.
Step-by-step explanation:
The muscle in the deep posterior compartment that is responsible for controlling pronation of the forearm is the tibialis posterior. This muscle originates from the interosseous membrane and adjacent bones and inserts into the tarsal and metatarsal bones of the foot. Its main functions are to invert and plantarflex the foot, as well as help maintain the medial longitudinal arch of the foot.
However, if we're referring to the deep posterior compartment of the forearm which controls pronation, this would typically involve the muscles associated with pronation such as the pronator teres and pronator quadratus, although they are not traditionally listed under the deep posterior compartment of the forearm.