For any quantum state defined with a continuous position, the Wigner function is a quasiprobability distribution on phase space. It has many properties, such as that its marginal are probability distributions, although the function itself can be negative.
Usually, people refer to the Wigner function (or any of several related function such as the Q-function, the Husimi-function, etc.) in the way that they have a quantum state and ask about the properties of its Wigner functions. Or, given Wigner functions with certain properties, what are the corresponding properties of their quantum states.
Q: I'm interested in the other direction. Given an arbitrary function from phase-space, what are necessary and sufficient conditions for this function to be the Wigner function of a quantum state?
Certainly, all mathematical properties listed in the Wikipedia article linked to above are necessary, although some are probably redundant. But are they sufficient? This paper "Wigner functions and Weyl transforms for pedestrians" also gives certain properties, but I could not find an answer to my question, neither there, nor anywhere else.