Naive deformation quantisation of a symplectic manifold

Question

Consider a $2n$-dimensional symplectic manifold $(M,\omega)$ with an open cover $(U_i)$ such that $U_i$ is symplectomorphic to $T^{*}V_i$ of some non-symplectic manifold $V_i$ for all $i$. Since each cotangent bundle $T^*V_i$ is quantised by the sheaf of differential operators $\pi^*\mathcal{D}_{V_i}$, where $\pi: T^*V_i\rightarrow V_i$, we should, at least hypothetically, be able to construct a formal quantisation of $M$ by gluing all the sheaves $\pi^*\mathcal{D}_{V_i}$. I am interested if this has been done in the literature and if not why not? I would appreciate all suggestions and comments.