Anomalies are a powerful way to gain insight into possible lattice regularizations of a quantum field theory. In this work, we consider lattice regularizations of a class of the toy-models of QCD: the 1+1-dimensional asymptotically-free Grassmanian nonlinear sigma models with a theta term. We argue that the continuum anomaly for a given symmetry can be matched by a manifestly symmetric lattice regularization only if (i) the symmetry action is offsite, or (ii) if the continuum anomaly is reproduced exactly on the lattice. Using the Grassmanian nonlinear sigma models as a case study, we provide examples of lattice regularizations in which both possibilities are realized. For possibility (i), we generalize recent work for the  O(3)  NLSM with an arbitrary theta term, where it was regulated using model of qubits with a small extra dimension, solving a sign problem present in conventional formulations of theta vacua. We argue that Grassmannian NLSM can be obtained similarly from SU(N) antiferromagnets with a well-defined continuum limit, reproducing both the IR physics of theta vacua and the UV 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 Luscher reproduces the anomaly exactly on the lattice, providing a realization of the second possibility.