Characterizing relative decidability in terms of model completeness

A theory $T$ is said to be relatively decidable if for every model of $T$, one can compute the elementary diagram of that model from its atomic diagram together with $T$. We verify a conjecture of Chubb, Miller, and Solomon by showing that for complete theories $T$, $T$ is relatively decidable if and only if $T$ has a conservative model complete extension of the form $T\cup \{\phi (\bar{c})\}$ where $T\models \mathrm{\exists }\bar{x}\phi (\bar{x})$. We also show that no such characterization works for incomplete theories.