  Anomalies are a powerful way to gain insight into possible lattice
regularizations of a quantum field theory. In this work, we argue that the
continuum anomaly for a given symmetry can be matched by a
manifestly-symmetric, local, lattice regularization in the same spacetime
dimensionality only if (i) the symmetry action is offsite, or (ii) if the
continuum anomaly is reproduced exactly on the lattice. We consider lattice
regularizations of a class of prototype models of QCD: the (1+1)-dimensional
asymptotically-free Grassmannian nonlinear sigma models (NLSMs) with a $\theta$
term. Using the Grassmannian NLSMs as a case study, we provide examples of
lattice regularizations in which both possibilities are realized. For
possibility (i), we argue that Grassmannian NLSMs can be obtained from
$\mathrm{SU}(N)$ antiferromagnets with a well-defined continuum limit,
reproducing both the infrared physics of $\theta$ vacua and the ultraviolet
physics of asymptotic freedom. These results enable the application of new
classical algorithms to lattice Monte Carlo studies of these quantum field
theories, and provide a viable realization suited for their quantum simulation.
On the other hand, we show that, perhaps surprisingly, the conventional lattice
regularization of $\theta$ vacua due to Berg and L\"uscher reproduces the
anomaly exactly on the lattice, providing a realization of the second
possibility.
