  We investigate from first principles the proposed destruction of the
controversial eg' pockets in the Fermi surface of NaxCoO2 due to Na disorder,
by calculating its k-dependent configurational averaged spectral function
<A(k,w)>. To this end, a Wannier function based method is developed that treats
the effects of disorder beyond the mean field. Remarkable spectral broadenings
of order ~eV are found for the oxygen orbitals, possibly explaining their
absence in the experiments. Contrary to the current lore, however, the eg'
pockets remain almost perfectly coherent. The developed method is expected to
generate exciting opportunities in the study of the countless functional
materials that owe their important electronic properties to disordered dopants.
