We consider semiclassical gravity with a non-minimally coupled Klein-Gordon field, with mass term $m^2 \geq 0$ and curvature coupling $\xi = 1/2$. We show that there is a class of Hadamard two-point functions for which the semiclassical Einstein equations coupled to the Klein-Gordon field take the form of a well-posed hyperbolic quasi-linear system. This class defines a set of quasi-free states, subject to the positivity condition, for which (the state-dependent part of) the expectation value of the renormalised stress-energy tensor is proportional to the spacetime metric. The case in which the proportionality constant vanishes is a quantum version of the so-called gravitational Cheshire effect.
