  We report a comprehensive ARPES study of NaFeAs, a prototypical parent
compound of the Fe-based superconductors. By mechanically detwinning the
samples, we show that in the nematic phase (below the structural transition at
$T_s$ = 54 K but above the antiferromagnetic transition at $T_N$ = 43 K)
spectral weight is detected on only the elliptical electron pocket along the
longer $a_{orth}$ axis. This dramatic anisotropy is likely to arise as a result
of coupling to a fluctuating antiferromagnetic order in the nematic phase. In
the long-range ordered antiferromagnetic state below $T_N$, this single
electron pocket is backfolded and hybridises with the hole bands, leading to
the reconstructed Fermi surface. By careful analysis of the $k_z$ variation, we
show that the backfolding of spectral weight in the magnetic phase has a
wavector of ($\pi$,0,$\pi$), with the $c$-axis component being in agreement
with the magnetic ordering in NaFeAs observed by neutron scattering. Our
results clarify the origin of the tiny Fermi surfaces of NaFeAs at low
temperatures and highlight the importance of the three-dimensional aspects of
the electronic and magnetic properties of Fe-based superconductors.
