In the MathOverflow question "Are the two meanings of 'undecidable' related" and its answers, connections are drawn between undecidability of a statement over a theory and undecidability of a decision problem by Turing machines. For example Joel David Hamkins's answer shows that for a $\Sigma_1$ sound, computably axiomatizable theory $T$, for any Turing-undecidable decision problem $A\subset\mathbb N$, there are infinitely many $n$ such that "$n\in A$?" is undecidable in $T$.
However, are the two meanings of "complete" related? Specifically are there any results connecting computability-theory completeness to a set relevant to the notion of proof-theory completeness?