  We study axion effective field theories (EFTs), with a focus on axion
couplings to massive chiral gauge fields. We investigate the EFT interactions
that participate in processes with an axion and two gauge bosons, and we show
that, when massive chiral gauge fields are present, such interactions do not
entirely originate from the usual anomalous EFT terms. We illustrate this both
at the EFT level and by matching to UV-complete theories. In order to assess
the consistency of the Peccei--Quinn (PQ) anomaly matching, it is useful to
introduce an auxiliary, non-dynamical gauge field associated to the PQ
symmetry. When applied to the case of the Standard Model (SM) electroweak
sector, our results imply that anomaly-based sum rules between EFT interactions
are violated when chiral matter is integrated out, which constitutes a smoking
gun of the latter. As an illustration, we study a UV-complete chiral extension
of the SM, containing an axion arising from an extended Higgs sector and heavy
fermionic matter that obtains most of its mass by coupling to the Higgs
doublets. We assess the viability of such a SM extension through electroweak
precision tests, bounds on Higgs rates and direct searches for heavy charged
matter. At energies below the mass of the new chiral fermions, the model
matches onto an EFT where the electroweak gauge symmetry is non-linearly
realised.
