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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.