Answer by Henry Towsner for Notable examples of syntactic proofs whose existence is guaranteed by completeness, but having been found later than a semantic proof?

There's a good reason that concrete examples are going to be rare. Existence of a first-order proof (at least in a countable language) is a $\Sigma_1$ property in the language of arithmetic, and it's a general metamathematical principle that proofs of $\Sigma_1$ statements should be constructive.

More precisely, if you have a proof that such a proof exists via the completeness theorem, one should be able to formalize this proof in some strong enough theory of arithmetic and then extract a syntactic proof (for instance, using the functional (or "Dialectica") interpretation). That doesn't mean there can't be such examples---the process of extracting a syntactic proof could be tedious, or simply not done yet---but it's unusual and unlikely to last for long if there's interest in obtaining a syntactic proof.