For a complex manifold $X$, there exists a ring of formal differential operators on $T^* M$, normally denoted as $\mathscr{E}_X$. Locally, a section of $\mathscr{E}_X$ is given by a formal sum $$P = \sum_{-\infty \leq j \leq m} p_j$$ where $p_j \in \mathcal{O}_{T^* X}(j)$, holomorphic functions on $T^* X$ of degree $j$ on the cotangent direction, satisfying certain growth conditions, and its product is dictated by how principal symbols multiply. (For details, see for example Schapira's book.) One main property of $\mathscr{E}_X$ is the it microlocalizes $\mathscr{D}_X$ in the sense that $\pi^{-1}\mathscr{D}_X \hookrightarrow \mathscr{E}_X$ is faithfully flat.
Now, in the algebraic setting, $\mathscr{D}_X$-modules can be identified with (quasi/in)-coherent sheaves on the De Rham space $X_{dR}$ where $X$ is a smooth scheme as discussed in Gaitsgory and Rozenblyum's book. (It might be true for the analytic setting, in some correct sense, but I do not know a reference.)
My question is if there is an interpretation of $\mathscr{E}_X$-modules as some version of coherent sheaves on some stack naturally associated to $X$ in either the algebraic or analytic setting? And if so, is there a reference?