  Halide perovskites have emerged as promising photonic materials for
fundamental physics studies and technological applications. Their potential for
nonlinear optics has also drawn great interest recently; yet, to date,
continuous-wave (CW) nonlinearities have remained elusive. Here we demonstrate
CW nonlinear phenomena in a CsPbBr$_3$ perovskite cavity. We first demonstrate
optical bistability -- the hallmark of single-mode coherent nonlinear optics.
Next we exploit the interplay of nonlinearity and birefringence to demonstrate
nonlinear control over the polarization of light. Finally, by measuring the
optical hysteresis of our cavity as a function of temperature, we find a
dramatic enhancement of the nonlinearity around 65 K. This enhancement is
indicative of a phase transition in CsPbBr$_3$. Our results position CsPbBr$_3$
cavities as an exceptional platform for nonlinear optics, offering strong CW
nonlinearity and birefringence which are furthermore tunable. In addition, our
approach to uncover a phase transition of matter via optical hysteresis
measurements is promising for exploring strongly correlated states of
light-matter systems.
