  We propose a novel mechanism for a nonequilibrium phase transition in a
$U(1)$-broken phase of an electron-hole-photon system, from a Bose-Einstein
condensate of polaritons to a photon laser, induced by the non-Hermitian nature
of the condensate. We show that a (uniform) steady state of the condensate can
always be classified into two types, namely, arising either from lower or
upper-branch polaritons. We prove (for a general model) and demonstrate (for a
particular model of polaritons) that an exceptional point where the two types
coalesce marks the endpoint of a first-order-like phase boundary between the
two types, similar to a critical point in a liquid-gas phase transition. Since
the phase transition found in this paper is not in general triggered by
population inversion, our result implies that the second threshold observed in
experiments is not necessarily a strong-to-weak-coupling transition, contrary
to the widely-believed understanding. Although our calculation mainly aims to
clarify polariton physics, our discussion is applicable to general
driven-dissipative condensates composed of two complex fields.
