= 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 $T\models\theta\to\varphi$ then makes $a$ satisfy every member of the type. Hence
$$
\boxed{\text{Every principal type is realized in every model of }T.}
$$
Completeness is what turns possible realization of the isolating formula into realization in every model.
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