  Recent measurements of the cosmic X-ray and radio backgrounds (CXB/CRB,
respectively) obtained with Chandra and ARCADE2 report signals in excess of
those expected from known sources, suggesting the presence of a yet
undiscovered population of emitters. We investigate the hypothesis that such
excesses are due to primordial black holes (PBHs) which may constitute a
substantial fraction of dark matter (DM). We present a novel semi-analytical
model which predicts X-ray and radio emission due to gas accretion onto PBHs,
assuming that they are distributed both inside DM halos and in the
intergalactic medium (IGM). Our model includes a self-consistent treatment of
heating/ionization feedback on the surrounding environment. We find that (i)
the emission from PBHs accreting in the IGM is subdominant at all times ($1\%
\leq I_{\rm IGM}/I_{\rm tot} \leq 40\% $); (ii) most of the CXB/CRB emission
comes from PBHs in DM mini-halos ($M_h \leq 10^6\ M_{\odot}$) at early epochs
($z>6$). While a small fraction ($f_{\rm PBH} \simeq 0.3\%$) of DM in the form
of PBHs can account for the total observed CXB excess, the CRB one cannot be
explained by PBHs. Our results set the strongest existing constraint on $
f_{\rm PBH} \leq 3\times 10^{-4}\ (30/M_{\rm PBH})$ in the mass range $1-1000\,
M_\odot$. Finally, we comment on the implications of our results on the global
$\rm H_I$ 21cm signal.
