Utilizing molecular dynamics simulations, we report a non-monotonic dependence of the shear stress on the strength of an external magnetic eld (H) in a liquid-crystalline mixture of magnetic and non-magnetic anisotropic particles.This non-monotonic behavior is in sharp contrast with the well-studied monotonic H-dependency of the shear stress in conventional ferro uids, where the shear stress increases with H until it reaches a saturation value. We relatethe origin of this non-monotonicity to the competing eects of particle alignment along the shear-induced direction, on the one hand, and the magnetic eld direction on the other hand. To isolate the role of these competing eects,we consider a two-component mixture composed of particles with eectively identical steric interactions, where the orientations of a small fraction, i.e. the magnetic ones, are coupled to the external magnetic eld. By increasing Hfrom zero, the orientations of the magnetic particles show a Freederickz-like transition and eventually start deviating from the shear-induced orientation, leading to an increase in shear stress. Upon further increase of H, a demixingof the magnetic particles, from the non-magnetic ones, occurs which leads to a drop in shear stress, hence creating a non-monotonic response to H. Unlike the equilibrium demixing phenomena reported in previous studies, the demixingobserved here is neither due to size-polydispersity nor due to a wall-induced nematic transition. Based on a simplied Onsager analysis, we rather argue that it occurs solely due to packing entropy of particles with dierent shear- or magnetic-eld-induced orientations.