  A gauge-invariant wave equation for the dynamics of hybrid quantum-classical
systems is formulated by combining the variational setting of Lagrangian paths
in continuum theories with Koopman wavefunctions in classical mechanics. We
identify gauge transformations with unobservable phase factors in the classical
phase-space and we introduce gauge invariance in the variational principle
underlying a hybrid wave equation previously proposed by the authors. While the
original construction ensures a positive-definite quantum density matrix, the
present model also guarantees the same property for the classical Liouville
density. After a suitable wavefunction factorization, gauge invariance is
achieved by resorting to the classical Lagrangian paths made available by the
Madelung transform of Koopman wavefunctions. Due to the appearance of a
phase-space analogue of the Berry connection, the new hybrid wave equation is
highly nonlinear and it is proposed here as a platform for further developments
in quantum-classical dynamics. Indeed, the associated model is Hamiltonian and
appears to be the first to ensure a series of consistency properties beyond
positivity of quantum and classical densities. For example, the model possesses
a quantum-classical Poincar\'e integral invariant and its special cases include
both the mean-field model and the Ehrenfest model from chemical physics.
