  In recent theoretical and experimental investigations, researchers have
linked the low-energy field theory of a Weyl semimetal gapped with a
charge-density wave (CDW) to high-energy theories with axion electrodynamics.
However, it remains an open question whether a lattice regularization of the
dynamical Weyl-CDW is in fact a single-particle axion insulator (AXI). In this
Letter, we use analytic and numerical methods to study both
lattice-commensurate and incommensurate minimal (magnetic) Weyl-CDW phases in
the mean-field state. We observe that, as previously predicted from field
theory, the two inversion- ($\mathcal{I}$-) symmetric Weyl-CDWs with $\phi =
0,\pi$ differ by a topological axion angle $\delta\theta_{\phi}=\pi$. However,
we crucially discover that $neither$ of the minimal Weyl-CDW phases at
$\phi=0,\pi$ is individually an AXI; they are instead quantum anomalous Hall
(QAH) and "obstructed" QAH insulators that differ by a fractional translation
in the modulated cell, analogous to the two phases of the Su-Schrieffer-Heeger
model of polyacetylene. Using symmetry indicators of band topology and
non-abelian Berry phase, we demonstrate that our results generalize to
multi-band systems with only two Weyl fermions, establishing that minimal
Weyl-CDWs unavoidably carry nontrivial Chern numbers that prevent the
observation of a static magnetoelectric response. We discuss the experimental
implications of our findings, and provide models and analysis generalizing our
results to nonmagnetic Weyl- and Dirac-CDWs.
