  We investigate the phase diagram of the complex cubic unitary ensemble of
random matrices with the potential $V(M)=-\frac{1}{3}M^3+tM$ where $t$ is a
complex parameter. As proven in our previous paper, the whole phase space of
the model, $t\in\mathbb C$, is partitioned into two phase regions,
$O_{\mathsf{one-cut}}$ and $O_{\mathsf{two-cut}}$, such that in
$O_{\mathsf{one-cut}}$ the equilibrium measure is supported by one Jordan arc
(cut) and in $O_{\mathsf{two-cut}}$ by two cuts. The regions
$O_{\mathsf{one-cut}}$ and $O_{\mathsf{two-cut}}$ are separated by critical
curves, which can be calculated in terms of critical trajectories of an
auxiliary quadratic differential. In our previous work the one-cut phase region
was investigated in detail. In the present paper we investigate the two-cut
region. We prove that in the two-cut region the endpoints of the cuts are
analytic functions of the real and imaginary parts of the parameter $t$, but
not of the parameter $t$ itself. We also obtain the semiclassical asymptotics
of the orthogonal polynomials associated with the ensemble of random matrices
and their recurrence coefficients. The proofs are based on the Riemann--Hilbert
approach to semiclassical asymptotics of the orthogonal polynomials and the
theory of $S$-curves and quadratic differentials.
