In this paper, we investigate a complex coupled dynamical system consisting of an unsteady Bingham-type fluid subject to nonmonotone frictional slip boundary conditions, an evolution equation for the temperature field, and a Hilfer fractional reaction–diffusion equation governing the concentration field. The primary objective is to establish the existence of solutions to the coupled system. The problem is reformulated in a variational form, yielding a coupled framework composed of a variational-hemivariational inequality and two quasilinear evolution equations, one of which is of fractional order. Under general assumptions, we prove the existence of weak solutions. The analysis relies primarily on the Rothe method, a time-discretization technique used to construct approximate solutions. A surjectivity theorem, together with tools from nonsmooth analysis, is then employed to establish solvability of the approximate problems and to derive the necessary a priori estimates. Finally, a limit passage argument completes the proof of the existence theorem for a weak solution to the original system. This work establishes a local existence result for this specific coupled system under the given assumptions. The result should be viewed as a first step toward a more complete analysis, such as uniqueness, global existence and regularity.