  Mixed iodine-bromine perovskites used in solar cells undergo below a critical
temperature an intrinsic demixing into phases with different iodine-bromine
compositions. In addition, under illumination they show nucleation of an
iodine-rich phase. We predict from thermodynamic considerations that in mixed
iodine-bromine perovskites like MAPb(I$_{1-x}$Br$_x$)$_3$ the interplay of
these effects can lead to coexistence of a bromine-rich, iodine-rich, and
nearly iodine-pure nucleated phase. This three-phase coexistence occurs in a
region in the composition-temperature phase diagram near the critical point for
intrinsic demixing. We investigate the hysteresis in the evolution of this
coexistence when temperature or illumination intensity are cycled. Depending on
the particular way the coexistence is established, nearly iodine-pure nuclei
should form either in the iodine-rich phase only or both in the bromine-rich
and iodine-rich phases. Experimental verification of this fundamentally novel
type of light-tunable three-phase coexistence should be possible by a
combination of absorption and photoluminescence experiments.
