  The possibility that a classical space-time and quantum matter cohabit at the
deepest level, i.e. the possibility of having a fundamental and not
phenomenological semiclassical gravity, is often disregarded for lack of a good
candidate theory. The standard semiclassical theory suffers from fundamental
inconsistencies (e.g.: Schr\"odinger cat sources, faster-than-light
communication and violation of the Born rule) which can only be ignored in
simple typical situations. We harness the power of spontaneous localization
models, historically constructed to solve the measurement problem in quantum
mechanics, to build a consistent theory of (stochastic) semiclassical gravity
in the Newtonian limit. Our model makes quantitative and potentially testable
predictions: we recover the Newtonian pair potential up to a short distance
cut-off (hence we predict no 1 particle self-interaction) and uncover an
additional gravitational decoherence term which depends on the specifics of the
underlying spontaneous localization model considered. We hint at a possible
program to go past the Newtonian limit, towards a consistent general
relativistic semiclassical gravity.
