 Solution

Let $\theta (x)$ isolate the principal type. Consistency with $T$ means $T\cup \{\exists x\theta (x)\}$ is satisfiable. Because $T$ is a complete theory, it must imply $\exists x\theta (x)$; the alternative would make that extension inconsistent.

Therefore every model of $T$ has a tuple $a$ satisfying $\theta$. Each implication $T\models \theta \to \phi$ then makes $a$ satisfy every member of the type. HenceCompleteness is what turns possible realization of the isolating formula into realization in every model.