The Wigner-Weyl transform $\mathfrak{W}$ is a bijective mapping between functions on a phase space and Hilbert space operators in order to map quantum mechanics into a phase-space formulation. Then the $\star$-product can be defined to be the morphism operation $$\mathfrak{W}(fg) = \mathfrak{W}(f) \star \mathfrak{W}(g).$$ Now I´m wondering about the most general underlying space where a Wigner-Weyl transform exists?
I know that Kontsevich (2003) has shown a $\star$-product for any finite-dimensional Poisson manifold, yet with an intrinsic definition of the $\star$-product. Does this already imply a Wigner-Weyl transform on any finite-dimensional Poisson manifold and how does it look like explicitly? Moreover, is this already the most general space where it exists?