  Microscopic thermal machines promise to play an important role in future
quantum technologies. Making such devices widely applicable will require
effective strategies to channel their output into easily accessible storage
systems like classical degrees of freedom. Here, we develop a self-consistent
theoretical framework that makes it possible to model such quantum-classical
hybrid devices in a thermodynamically consistent manner. Our approach is based
on the assumption that the quantum part of the device is subject to strong
decoherence and dissipation induced by a thermal reservoir. Due to the ensuing
separation of time scales between slowly evolving classical and fast relaxing
quantum degrees of freedom, the dynamics of the hybrid system can be described
by means of adiabatic-response theory. We show that, upon including
fluctuations in a minimally consistent way, the resulting equations of motion
can be equipped with a first and second law, both on the ensemble level and on
the level of individual trajectories of the classical part of the system, where
thermodynamic quantities like heat and work become stochastic variables. As an
application of our theory, we work out a physically transparent model of a
quantum-classical hybrid engine, whose working system consists of a chain of
Rydberg atoms, which is confined in an optical cavity and driven by periodic
temperature variations. We show by means of numerical simulations that the
engine can sustain periodic oscillations of a movable mirror, which acts as a
classical load, against external friction and extract the full distributions of
input heat and output work. By making the statistics of thermodynamic processes
in quantum-classical hybrid systems accessible without the need to further
specify a measurement protocol, our work contributes towards bridging the
long-standing gap between classical and quantum stochastic thermodynamics.
