  Upon revisiting the Hamiltonian structure of classical wavefunctions in
Koopman-von Neumann theory, this paper addresses the long-standing problem of
formulating a dynamical theory of classical-quantum coupling. The proposed
model not only describes the influence of a classical system onto a quantum
one, but also the reverse effect -- the quantum backreaction. These
interactions are described by a new Hamiltonian wave equation overcoming
shortcomings of currently employed models. For example, the density matrix of
the quantum subsystem is always positive-definite. While the Liouville density
of the classical subsystem is generally allowed to be unsigned, its sign is
shown to be preserved in time for a specific infinite family of hybrid
classical-quantum systems. The proposed description is illustrated and compared
with previous theories using the exactly solvable model of a degenerate
two-level quantum system coupled to a classical harmonic oscillator.
