  We extend our previous work (Proukakis {\em et al.}, Phys.~Rev.~D~108,~083513
(2023)) on the dynamics of bosonic, non-relativistic and self-interacting dark
matter that simultaneously contains both a ``fuzzy'' low-momentum component and
one with higher momenta that may be well approximated as a collection of
distinct particles and described by a corresponding phase-space distribution.
Starting from the non-relativistic Schwinger-Keldysh action and working beyond
leading-order in the Keldysh basis fields, encoding stochastic fluctuations of
the slow modes and all fluctuations of the fast modes, we obtain stochastic
self-consistently coupled Gross-Pitaevskii, collisional Boltzmann kinetic and
Poisson equations. Our final set of equations, which feature various
collisional (dissipative and scattering) contributions and two corresponding
independent stochastic force terms, are consistent with generalized
fluctuation-dissipation type relations in the limit of thermal equilibrium
between the particles.
