  Lorentz invariance plays a fundamental role in modern physics. However, tiny
violations of the Lorentz invariance may arise in some candidate quantum
gravity theories. Prominent signatures of the gravitational Lorentz invariance
violation (gLIV) include anisotropy, dispersion, and birefringence in the
dispersion relation of gravitational waves (GWs). Using a total of 50 GW events
in the GW transient catalogs GWTC-1 and GWTC-2, we perform an analysis on the
anisotropic birefringence phenomenon. The use of multiple events allows us to
completely break the degeneracy among gLIV coefficients and globally constrain
the coefficient space. Compared to previous results at mass dimensions 5 and 6
for the Lorentz-violating operators, we tighten the global limits of 34
coefficients by factors ranging from $2$ to $7$.
