  We study a class of theories in which space-time is treated classically,
while interacting with quantum fields. These circumvent various no-go theorems
and the pathologies of semi-classical gravity, by being linear in the density
matrix and phase-space density. The theory can either be considered fundamental
or as an effective theory where the classical limit is taken of space-time. The
theories have the dynamics of general relativity as their classical limit and
provide a way to study the back-action of quantum fields on the space-time
metric. The theory is invariant under spatial diffeomorphisms, and here, we
provide a methodology to derive the constraint equations of such a theory by
imposing invariance of the dynamics under time-reparametrization invariance.
This leads to generalisations of the Hamiltonian and momentum constraints. We
compute the constraint algebra for a wide class of realisations of the theory
(the "discrete class") in the case of a quantum scalar field interacting with
gravity. We find that the algebra doesn't close without additional constraints,
although these do not necessarily reduce the number of local degrees of
freedom.
