  We study the decay properties of Wigner kernels for Fourier integral
operators of types I and II. The symbol spaces that allow a nice decay of these
kernels are the Shubin classes $\Gamma^m(\mathbb{R^{2d}})$, with negative order
$m$. The phases considered are the so-called tame ones, which appear in the
Schr\"odinger propagators. The related canonical transformations are allowed to
be nonlinear. It is the nonlinearity of these transformations that are the main
obstacles for nice kernel localizations when symbols are taken in the
H\"ormander's class $S^{0}_{0,0}(\mathbb{R^{2d}})$. Here we prove that Shubin
classes overcome this problem and allow a nice kernel localization, which
improves with the decreasing of the order $m$.
