<p>In this paper, we devote our attention to exploring a new class of dynamical systems composed of the Navier-Stokes equations for a Bingham fluid with a nonmonotone friction-type slip boundary condition and a Hilfer fractional reaction-diffusion equation with a Neumann boundary condition. The system couples a variational-hemivariational inequality and a fractional quasilinear evolution equation. First, under appropriate conditions, the solvability to the coupled system is established by the Rothe technique which is based on the time semidiscrete approximation backward Euler method and a feedback iteration approach. Next, we employ a surjectivity theorem and tools from nonsmooth analysis to prove the existence of a solution to the approximation problem and to derive a priori estimates. Finally, the limit process is carried out to establish the existence result of a weak solution of the system.</p>