  We examine the role of the $U(1)_A$ anomaly in a parity doublet model of
nucleons which include the chiral variant and invariant masses. Our model
expresses the $U(1)_A$ anomaly by the Kobayashi-Maskawa-'t\,Hooft (KMT)
interaction in the mesonic sector. After examining the roles of the KMT term in
vacuum, we discuss its impacts on nuclear equations of state (EOS). The
$U(1)_A$ anomaly increases the masses of the $\eta'$ and $\sigma$ mesons and
enhances the chiral symmetry breaking. The $U(1)_A$ anomaly enlarges the energy
difference between chiral symmetric and symmetry broken vacuum; in turn, the
chiral restoration at high density adds a larger energy density (often referred
as a bag constant) to EOSs than in the case without the anomaly, leading to
softer EOSs. Including these $U(1)_A$ effects, we update the previously
constructed unified equations of state that interpolate the nucleonic EOS at
$n_B \le 2n_0$ ($n_{0} = 0.16\, \rm{fm^{-3}}$: nuclear saturation density) and
quark EOS at $n_B \ge 5n_0$. The unified EOS is confronted with the
observational constraints on the masses and radii of neutron stars. The
softening of EOSs associated with the $U(1)$ anomaly reduces the overall radii,
relaxing the previous constraint on the chiral invariant mass $m_0$. Including
the attractive nonlinear $\rho$-$\omega$ coupling to improved estimates for the
slope parameter in the symmetry energy, our new estimate is $400\,{\rm MeV}
\leq m_0 \leq 700\,{\rm MeV}$, with $m_0$ smaller than our previous estimate by
$\sim 200$ MeV.
