  Axions and other pseudoscalar fields comprise an interesting class of
ultralight dark matter candidates, that may independently play a role in
solving the strong $CP$ problem. In the presence of $CP$-violating sources,
these pseudoscalar fields can develop a coherent non-derivative coupling to
nucleons, $\bar g_{aNN}$, thus mediating `mass-mass' and `mass-spin' forces in
matter that can be probed experimentally. We revisit the non-perturbative
generation of these $CP$-odd axion forces, and refine estimates of $\bar
g_{aNN}$ generated by the EDMs and color EDMs of quarks. We also revisit the
Standard Model contribution to $CP$-odd axion couplings generated by the phase
of the Cabibbo-Kobayashi-Maskawa quark mixing matrix.
