  We analyze the dynamics and observational signatures of axion clouds formed
via the superradiant instability around primordial black holes, focusing on the
mass range $10^{14}-10^{18}$ kg where the latter may account for all the dark
matter. We take into account the leading effects of axion self-interactions,
showing that, even though these limit the number of axions produced within each
cloud, a large number of superradiant axions become free of the black hole's
gravitational potential and accumulate in the intergalactic medium or even in
the host galaxy, depending on their escape velocity. This means that primordial
black hole dark matter may lead to a sizeable astrophysical population of
non-relativistic axions, with masses ranging from 0.1 eV to 1 MeV, depending on
the primordial black hole mass and spin. We then show that if such axions
couple to photons their contribution to the galactic and extragalactic
background flux, mainly in the X-ray and gamma-ray band of the spectrum, is
already beyond current observational limits for a large range of parameters
that are, therefore, excluded. We finish by showing the prospects of the Athena
X-ray telescope to further probe this co-existence of primordial black holes
and axions.
