In this paper, we prove the local uniqueness of an inverse problem arising in the nonstationary flow of a nonhomogeneous incompressible asymmetric fluid in a bounded domain with a smooth boundary. The direct problem is an initial boundary value problem for a system where the unknowns are the velocity field of the fluid particles, the angular velocity of rotation of the fluid particles, the mass density of the fluid, and the pressure distribution. The inverse problem consists of recovering external forces to the linear and angular momentum equations by assuming a set of measurements as an integral overspecified condition. We introduce and prove several a priori estimates. We characterize the inverse problem solutions using an operator equation of a second kind, deduced from applying the Helmholtz decomposition. We prove several properties of the associated operator, which implies that the Tikhonov fixed point theorem hypothesis is valid. Then, we deduce the local unique solvability of the inverse problem.