Let $ZF$ be the Zermelo-Fraenkel set theory and let $AC_\omega$ be the countable axiom of choice.
In $ZF+AC\omega$ one can prove the propositional compactness theorem for countable languages.
My question is: in $ZF+AC_\omega$ can I prove the propositional compactness theorem for arbitrary languages?
I suppose the answer is negative, so, if I assume $ZF+AC_\omega+$"propositional compactness theorem for arbitrary languages" can I obtain a contradiction?
The answer to the first question is negative, even if you strengthen $AC_\omega$ to $DC$ (dependent choice). Solovay's model for "all sets of reals are Lebesgue measurable" gives a counterexample, since there are no non-principal ultrafilters on $\omega$ in this model (and propositional compactness implies that all proper filters can be extended to ultrafilters). If you worry about the inaccessible needed for Solovay's model, then Shelah's model for "all sets of reals have the Baire property" works as well.
For the second question, note that the theory you asked about is included in ZFC, so it is consistent (provided of course that ZF is consistent).