  We establish a derived geometric Satake equivalence for the quaternionic
general linear group GL_n(H). By applying the real-symmetric correspondence for
affine Grassmannians, we obtain a derived geometric Satake equivalence for the
symmetric variety GL_2n/Sp_2n. We explain how these equivalences fit into the
general framework of a geometric Langlands correspondence for real groups and
the relative Langlands duality conjecture. As an application, we compute the
stalks of the IC-complexes for spherical orbit closures in the quaternionic
affine Grassmannian and the loop space of GL_2n/Sp_2n. We show the stalks are
given by the Kostka-Foulkes polynomials for GL_n but with all degrees doubled.
