In this Letter, we propose an original and novel experimental method to characterize both the onset and morphology of Taylor-Couette instability occurring in a non-Newtonian cylindrical Couette flow. Using synchrotron-based rheological small angle x-ray scattering experiments, we jointly investigate the shear-thinning behavior of natural swelling clays suspensions and the associated anisotropy developing in such media. Combined with a linear stability analysis for power law fluids, a destabilizing effect is shown both numerically and experimentally and the vortices morphology is found to be dependent on the fluid index. Still, the strong destabilizing effect and large vortex size can not be assigned to shear-thinning only, which clearly evidences the impact of medium anisotropy on Taylor-Couette instability.