In this work, we investigate the role of elastic inclusions in a plastically anisotropic matrix using crystal plasticity modeling and simulation. Magnesium (Mg) is taken as a model matrix material, which exhibits large plastic anisotropy that originates from slip and twinning mechanisms with dramatically different activation stresses. Using an idealized setup of a periodic unit cell comprising single crystal Mg matrix with an embedded elastic inclusion, we investigate the role of inclusion shape and alignment in the evolution of composite flow responses under compressive and tensile loads. This idealization serves as a model setup for highly textured microstructures that result from extrusion or rolling processes. Detailed analysis reveals how slip and twinning mechanisms evolve with strain and how they depend on the reinforcement morphology. Results indicate that under the loading condition that preferentially activate {101¯2} extension twinning, the inclusion morphology and alignment significantly influence the amount of flow hardening at a given strain and its evolution as a function of increasing strain. This twinning induced hardening effect exists in addition to the classical hydrostatic constraint effect induced by the presence of elastically stiff inclusions. We propose a simple empirical expression, which quantifies this coupling. On the other hand, when extension twinning is not an active deformation mechanism, the flow hardening characteristics are similar to the classical MMCs that deform by dislocation slip. We also investigate the effects of reinforcement aspect ratio and volume fraction on the composite responses. Finally, we briefly discuss how these observations on single crystal MMC models can be extended to textured polycrystalline Mg MMCs.