  We consider coincidence Reidemeister zeta functions for tame endomorphism
pairs of nilpotent groups of finite rank, shedding new light on the subject by
means of profinite completion techniques.
  In particular, we provide a closed formula for coincidence Reidemeister
numbers for iterations of endomorphism pairs of torsion-free nilpotent groups
of finite rank, based on a weak commutativity condition, which derives from
simultaneous triangularisability on abelian sections. Furthermore, we present
results in support of a P\'olya-Carlson dichotomy between rationality and a
natural boundary for the analytic behaviour of the zeta functions in question.
