  In this paper, we initiate the study of global stability for anisotropic
systems of quasilinear wave equations. Equations of this kind arise naturally
in the study of crystal optics, and they exhibit birefringence. We introduce a
physical space strategy based on bilinear energy estimates that allows us to
prove decay for the nonlinear problem. This uses decay for the homogeneous wave
equation as a black box. The proof also requires us to interface this strategy
with the vector field method and take advantage of the scaling vector field. A
careful analysis of the spacetime geometry of the interaction between waves is
necessary in the proof.
