  We prove that cyclic subgroup separability is preserved under exponential
completion for groups that belong to a class that includes all coherent RAAGs
and toral relatively hyperbolic groups; we do so by exploiting the structure of
these completions as iterated free products with commuting subgroups. From this
we deduce that the cyclic subgroups of limit groups over coherent RAAGs are
separable, answering a question of Casals-Ruiz, Duncan and Kazachov. We also
discuss relations between free products with commuting subgroups and the word
problem, and recover the fact that limit groups over coherent RAAGs and toral
relatively hyperbolic groups have a solvable word problem.
