There actually is a formulation, due to Torres-Vega and Frederick, but let me hasten to add that you really don't want to go there, for about half a dozen good reasons, most of them very technical, and some quite conceptual.
As @ACuriousMind already outlined, the classic satisfactory and powerful formulation of QM in phase space involves the analog of the density matrix (which is traced with observables to yield expectation values in Hilbert space), the Wigner Function (which is integrated with c-number observables in phase space instead). You might consider this introduction Quantum mechanics in phase space, by Curtright, Fairlie & me, if you wished to know more about what this formulation gets you that the other two, Hilbert space, and path integrals, don't. (You might choose to focus on eqn (91) and Exercise 0.11 there for a gateway to the T-V & F reformulation. Remember, all phases needed for interference are already in off-diagonal Wigner Functions!)
The Torres-Vega/Frederick formulation I mentioned finds its meaning by painful projection/reference to the Wigner Function mentioned, but, once you stretch your conceptual horizons to appreciate that formulation, you might realize you really have no use for the Schroedinger equation you are envisioning, but its density matrix analog, the von Neumann equation, instead, much as you derive stat mech expectation values by integrating agains Liouville densities.
Added edit: The Ambiguity function connects to the WF via a 2d Fourier transform, as indicated; basically, all roads go through the WF. For instance, by multiplying the coordinate to the momentum space wave functions, you came close to defining the Mehta distribution function, cf my answer to 233353, Exercise 0.19, but by a phase... what Terletsky and Blokhintsev did in the 30s. The systematic theory of all of these connexions could be accessed in L Cohen's book.