We introduce and study a system of coupled nonlocal nonlinear Schrödinger equations that interpolates between the mixed, focusing–defocusing Manakov system on one hand and a limiting case of the intermediate nonlinear Schrödinger equation on the other. We show that this new system, which we call the intermediate mixed Manakov (IMM) system, admits multi-soliton solutions governed by a complexification of the hyperbolic Calogero–Moser (CM) system. Furthermore, we introduce a spatially periodic version of the IMM system, for which our main result is a class of exact solutions governed by a complexified elliptic CM system.