  The original theory of quasi-metric gravity, admitting only a partial
coupling between space-time geometry and the active stress-energy tensor, is
too restricted to allow the existence of gravitational waves in vacuum.
Therefore, said theory can at best be regarded as a wave-less approximation
theory. However, the requirement that the weak-field limit of the contracted
Bianchi identities should be consistent with the Newtonian limit of the local
conservation laws, forbids a full coupling between space-time geometry and the
active stress-energy tensor. Nevertheless, in this paper it is shown how it is
possible to relax the restrictions on quasi-metric space-time geometry
sufficiently to avoid these problems. That is, the original quasi-metric field
equations can be extended with one extra field equation, without having said
full coupling and such that the contracted Bianchi identities have a sensible
Newtonian limit. For weak fields in vacuum, said extra field equation has a
dynamical structure somewhat similar to that of its counterpart in canonical
general relativity (GR). In this way, the prediction of weak GR-like
gravitational waves in vacuum becomes possible. Moreover, exact results from
the original quasi-metric gravitational theory are recovered for metrically
static systems and for isotropic cosmology. This means that the current
experimental status of the extended quasi-metric gravitational theory is the
same as for the original theory, except for the prediction of weak GR-like
gravitational waves in vacuum.
