  We study electroweak scale Dark Matter (DM) whose interactions with baryonic
matter are mediated by a heavy anomalous $Z'$. We emphasize that when the DM is
a Majorana particle, its low-velocity annihilations are dominated by loop
suppressed annihilations into the gauge bosons, rather than by p-wave or
chirally suppressed annihilations into the SM fermions. Because the $Z'$ is
anomalous, these kinds of DM models can be realized only as effective field
theories (EFTs) with a well-defined cutoff, where heavy spectator fermions
restore gauge invariance at high energies. We formulate these EFTs, estimate
their cutoff and properly take into account the effect of the Chern-Simons
terms one obtains after the spectator fermions are integrated out. We find
that, while for light DM collider and direct detection experiments usually
provide the strongest bounds, the bounds at higher masses are heavily dominated
by indirect detection experiments, due to strong annihilation into $W^+W^-$,
$ZZ$, $Z\gamma$ and possibly into $gg$ and $\gamma\gamma$. We emphasize that
these annihilation channels are generically significant because of the
structure of the EFT, and therefore these models are prone to strong indirect
detection constraints. Even though we focus on selected $Z'$ models for
illustrative purposes, our setup is completely generic and can be used for
analyzing the predictions of any anomalous $Z'$-mediated DM model with
arbitrary charges.
