  Understanding the evolution of carbon and iron in the Milky Way's halo is of
importance because these two elements play crucial roles constraining star
formation, Galactic assembly, and chemical evolution in the early Universe.
Here, we explore the spatial distributions of carbonicity, [C/Fe], and
metallicity, [Fe/H], of the halo system based on medium-resolution ($R \sim$
1,300) spectroscopy of $\sim$58,000 stars in the Southern Hemisphere from the
AAOmega Evolution of Galactic Structure (AEGIS) survey. The AEGIS carbonicity
map exhibits a positive gradient with distance, as similarly found for the
Sloan Digital Sky Survey (SDSS) carbonicity map of Lee et al. The metallicity
map confirms that [Fe/H] decreases with distance, from the inner halo to the
outer halo. We also explore the formation and chemical-evolution history of the
halo by considering the populations of carbon-enhanced metal-poor (CEMP) stars
present in the AEGIS sample. The cumulative and differential frequencies of
CEMP-no stars (as classified by their characteristically lower levels of
absolute carbon abundance, $A$(C) $\leq$ 7.1 for sub-giants and giants)
increases with decreasing metallicity, and is \textit{substantially higher than
previous determinations} for CEMP stars as a whole. In contrast, that of
CEMP-$s$ stars (with higher $A$(C)), remains almost flat, at a value
$\sim$10\%, in the range $-\,4.0 \lesssim$ [Fe/H] $\lesssim-$2.0. The
distinctly different behaviors of the CEMP-no and CEMP-$s$ stars relieve the
tension with population-synthesis models assuming a binary mass-transfer
origin, which previously struggled to account for the higher reported
frequencies of CEMP stars, taken as a whole, at low metallicity.
