  A $Z'$ gauge boson with mass around the (few) TeV scale is a popular example
of physics beyond the Standard Model (SM) and can be a fascinating remnant of a
Grand Unified Theory (GUT). Recently, $Z'$ models with non-universal couplings
to the SM fermions due to extra vector-like states have received attention as
potential explanations of the present $R_K$, $R_{K^{\ast}}$ anomalies; this
includes GUT model proposals based on the $\mathrm{SO}(10)$ group. In this
paper we further develop GUT models with a flavour non-universal low scale $Z'$
and clarify several outstanding issues within them. First, we successfully
incorporate a realistic neutrino sector (with linear and/or inverse low scale
seesaw mechanism), which was so far a missing ingredient. Second, we
investigate in detail their compatibility with the $R_K$, $R_{K^{\ast}}$
anomalies; we find that the anomalies do not have a consistent explanation
within such models. Third, we demonstrate that these models have other
compelling phenomenological features; we study the correlations between the
flavour violating processes of $\mu\to 3e$ and $\mu$-$e$ conversion in a muonic
atom, showing how a GUT imprint could manifest itself in experiments.
