  We extend our study of the large-$N$ expansion of general non-equilibrium
many-body systems with matrix degrees of freedom $M$, and its dual description
as a sum over surface topologies in a dual string theory, to the
Keldysh-rotated version of the Schwinger-Keldysh formalism. The Keldysh
rotation trades the original fields $M_\pm$ -- defined as the values of $M$ on
the forward and backward segments of the closed time contour -- for their
linear combinations $M_{\textrm{cl}}$ and $M_{\textrm{qu}}$, known as the
"classical" and "quantum" fields. First we develop a novel "signpost" notation
for non-equilibrium Feynman diagrams in the Keldysh-rotated form, which
simplifies the analysis considerably. Before the Keldysh rotation, each
worldsheet surface $\Sigma$ in the dual string theory expansion was found to
exhibit a triple decomposition into the parts $\Sigma^\pm$ corresponding to the
forward and backward segments of the closed time contour, and $\Sigma^\wedge$
which corresponds to the instant in time where the two segments meet. After the
Keldysh rotation, we find that the worldsheet surface $\Sigma$ of the dual
string theory undergoes a very different natural decomposition: $\Sigma$
consists of a "classical" part $\Sigma^{\textrm{cl}}$, and a "quantum
embellishment" part $\Sigma^{\textrm{qu}}$. We show that both parts of $\Sigma$
carry their own independent genus expansion. The non-equilibrium sum over
worldsheet topologies is naturally refined into a sum over the double
decomposition of each $\Sigma$ into its classical and quantum part. We apply
this picture to the classical limits of the quantum non-equilibrium system
(with or without interactions with a thermal bath), and find that in these
limits, the dual string perturbation theory expansion reduces to its
appropriately defined classical limit.
