  Motility-induced phase separation (MIPS), the phenomenon in which purely
repulsive active particles undergo a liquid-gas phase separation, is among the
simplest and most widely studied examples of a nonequilibrium phase transition.
Here, we show that states of MIPS coexistence are in fact only metastable for
three-dimensional active Brownian particles over a very broad range of
conditions, decaying at long times through an ordering transition we call
active crystallization. At an activity just above the MIPS critical point, the
liquid-gas binodal is superseded by the crystal-fluid coexistence curve, with
solid, liquid, and gas all coexisting at the triple point where the two curves
intersect. Nucleating an active crystal from a disordered fluid, however,
requires a rare fluctuation that exhibits the nearly close-packed density of
the solid phase. The corresponding barrier to crystallization is surmountable
on a feasible timescale only at high activity, and only at fluid densities near
maximal packing. The glassiness expected for such dense liquids at equilibrium
is strongly mitigated by active forces, so that the lifetime of liquid-gas
coexistence declines steadily with increasing activity, manifesting in
simulations as a facile spontaneous crystallization at extremely high activity.
