  We show that the soft X-ray excess in the Coma cluster can be explained by a
cosmic background of relativistic axions converting into photons in the cluster
magnetic field. We provide a detailed self-contained review of the cluster soft
X-ray excess, the proposed astrophysical explanations and the problems they
face, and explain how a 0.1-1 keV axion background naturally arises at
reheating in many string theory models of the early universe. We study the
morphology of the soft excess by numerically propagating axions through
stochastic, multi-scale magnetic field models that are consistent with
observations of Faraday rotation measures from Coma. By comparing to ROSAT
observations of the 0.2-0.4 keV soft excess, we find that the overall excess
luminosity is easily reproduced for $g_{a\gamma\gamma} \sim 2 \times 10^{-13}$
GeV$^{-1}$. The resulting morphology is highly sensitive to the magnetic field
power spectrum. For Gaussian magnetic field models, the observed soft excess
morphology prefers magnetic field spectra with most power in coherence lengths
on ${\cal O}(3 {\rm ~kpc})$ scales over those with most power on ${\cal O}(12
{\rm ~kpc})$ scales. Within this scenario, we bound the mean energy of the
axion background to $50\, {\rm eV}\lesssim \langle E_a \rangle \lesssim 250\,
{\rm eV}$, the axion mass to $m_a \lesssim 10^{-12}\,\hbox{eV}$, and derive a
lower bound on the axion-photon coupling $g_{a\gamma\gamma} \gtrsim
\sqrt{0.5/\Delta N_{\rm eff}}\, 1.4 \times 10^{-13}$ GeV$^{-1}$.
