  It has been recently claimed that the axion coupling to fermions is
responsible for an oscillating electric dipole moment (EDM) in the background
of axion dark matter. In this work, we re-examine the derivation of this
effect. Contrary to previous studies, we point out the physical relevance of an
axion boundary term, which is crucial in restoring the axion shift symmetry and
drastically affects the EDM phenomenology. To describe the latter, we introduce
the notion of a time-averaged effective axion EDM, which encodes the boundary
term and whose magnitude depends on the oscillation regime. For slow
oscillations, the boundary term washes out the standard oscillating EDM,
resulting in an exact cancellation in the static limit. Conversely, during fast
oscillations, the boundary term amplifies the effective EDM relatively to the
standard EDM contribution. This observable is especially interesting in the
case of the electron EDM. For an $\mathcal{O}(1)$ axion-electron coupling, the
overall size of the effective EDM in the regime of intermediate or fast
oscillations is comparable to the present static EDM limit.
