In a few MSE and MOL posts (for instance here and here ), it was mentioned that a characterization of sentences preserved under direct products was given by Weinstein in his PhD thesis. As noted there, it seems to be quite difficult to track down this PhD thesis, and hence to access the results contained.
I am particularly interested in a syntactic characterization of $\forall\exists$ Horn sentences. A naive idea would be that these should be the sentences preserved under both unions of chains and direct products (or perhaps reduced products), but it does not seem very obvious. It was mentioned above that Weinstein proved something to this effect.
Additionally, in case the above turns out to be true, I am interested whether this also holds for theories, i.e., if a theory is preserved under unions of chains and direct products, whether it could be axiomatised by universal existential Horn sentences.