also witnessed a rising interest in nano-mechanical, opto-mechanical and other types of hybrid quantum systems [@nosc; @nosc2; @nosc3; @nosc4; @nosc5; @nosc6; @nosc7; @nosc8; @nosc9; @nosc10; @nosc11; @nosc12; @nosc13; @nosc14; @nosc15; @nosc16; @nosc17; @nosc18; @nosc19; @nosc20; @nosc21; @nosc22; @nosc23]. Such systems often exhibit an interplay between classical and quantum effects, allowing them to be modeled by means of hybrid quantum-classical methods.

It has been known for a long time, that the dynamics and statistical mechanics of a quantum subsystem coupled to classical-like DOF can be formulated in terms of operator-valued quasi-probability functions in phase space [@silin; @rukhazade-silin; @balescu-1968; @zhang-balescu; @balescu-zhang; @osborn]. For example, the dynamics of nano-mechanical oscillators has been previously described by one of the authors in terms of operator-valued quasi-probability functions [@beck]. Such functions evolve through quasi-Lie brackets [@aleksandrov; @gerasimenko; @boucher-transchen; @gerasimenko2; @martens-fang; @prezhdo-kisil; @donoso-martens; @donoso-martens2; @qc-stat; @b3; @b4], which can also be augmented by dissipative terms when the energy is not conserved [@ray-barbados; @langevin-qc]. When the bath is described by canonically conjugate variables (and only in this case), a hybrid quantum-classical formalism may be derived. Starting from a fully quantum representation of the subsystem and bath DOF, one can perform a partial Wigner transform [@wigner] (over the bath 