  The present data on a number of observables in $b\to s\mu^+\mu^-$ processes
manifest some tensions with the standard model (SM). Assuming that these
anomalies have a new physics origin, we consider the possibility that a $Z'$
boson is responsible for them. We further assume that its interactions with
quarks also affect rare nonleptonic decays of the $\bar B_s^0$ meson which are
purely isospin-violating and tend to be dominated by electroweak-penguin
contributions, namely $\bar B_s^0\to(\eta,\eta',\phi)(\pi^0,\rho^0)$. Most of
these decays are not yet observed, and their rates are expected to be
relatively small in the SM. Taking into account constraints from various
measurements, including the evidence for $\bar B_s^0\to\phi\rho^0$ recently
seen by LHCb, we find that the $Z'$ effects on $\bar B_s^0\to(\eta,\phi)\pi^0$
can make their rates bigger than the SM predictions by up to an order of
magnitude. For $\bar B_s^0\to\eta'\pi^0,(\eta,\eta')\rho^0$, the enhancement
factors are at most a few. Since the $Z'$ contributions to the different
channels depend on different combinations of its couplings, observations of
more of these decays in future experiments, along with improved $b\to
s\mu^+\mu^-$ data, will probe this $Z'$ scenario more thoroughly.
