  The Axiom of Choice (AC for short) is the most (in)famous axiom of the usual
foundations of mathematics, ZFC set theory. The (non-)essential use of AC in
mathematics has been well-studied and thoroughly classified. Now, fragments of
countable AC not provable in ZF have recently been used in Kohlenbach's
higher-order Reverse Mathematics to obtain equivalences between closely related
compactness and local-global principles. We continue this study and show that
NCC, a weak choice principle provable in ZF and much weaker systems, suffices
for many of these results. In light of the intimate connection between Reverse
Mathematics and computability theory, we also study realisers for NCC, i.e.
functionals that produce the choice functions claimed to exist by the latter
from the other data. Our hubris of undertaking the hitherto underdeveloped
study of the computational properties of (choice functions from) AC leads to
interesting results. For instance, using Kleene's S1-S9 computation schemes, we
show that various total realisers for NCC compute Kleene's $\exists^3$, a
functional that gives rise to full second-order arithmetic, and vice versa. By
contrast, partial realisers for NCC should be much weaker, but establishing
this conjecture remains elusive. By way of catharsis, we show that the
Continuum Hypothesis (CH for short) is equivalent to the existence of a
countably based partial realiser for NCC. The latter kind of realiser does not
compute Kleene's $\exists^3$ and is therefore strictly weaker than a total one.
