  We consider semi-classical time evolution for the phase space Schr\"{o}dinger
equation. We construct a semi-classical phase space propagator in terms of
semi-classical wave packets by the Anisotropic Gaussian Approximation, related
to the Nearby Orbit Approximation. We deduce the canonical system in double
phase space, related to the Berezin-Shubin-Marinov canonical system and
construct a semi-classical asymptotic solution of the Cauchy problem for the
phase space Schr\"{o}dinger equation for initial WKB states. We illustrate the
method for sub-quadratic potentials in $\mathbb{R}$.
