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Undecidable completion of undecidable theory, and pairs of RCF

Given an undecidable collection of first-order sentences, is there necessarily a complete undecidable theory containing it? A direct attempt to prove it seems to require some control over the completions which need not exist, but I don't see a counterexample.

On a more concrete side, Macyntire proved that the theory of all pairs of real closed fields is undecidable. I am interested to know if there is a particular pair of real closed fields with undecidable theory.
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Undecidable completion of undecidable theory

Given an undecidable collection of first-order sentences, is there necessarily a complete undecidable theory containing it? A direct attempt to prove it seems to require some control over the completions which need not exist, but I don't see a counterexample.