I read this old question, and in particular I was interested in this answer.
Ostap says that for the concept of Cech-complete a generalization of the Baire's theorem holds, but he didn't say that a Cech-complete space is a Baire space, is this true? Furthermore is this theorem also a generalization of the theorem "a pseudocompact completely regular is a Baire space" (this is the exercise XI.10.10 of Dugundji)?
A Cech-complete space is indeed a Baire space (see Engelking General Topology ,Corr. 3.9.4) and any product of Cech-complete spaces is Baire (but not always Cech-complete). Something better holds for pseudocomplete spaces, introduced by Oxtoby (see this more recent paper for more info): they are Baire and all their products are still pseudocomplete (so still Baire). Pseudocompact completely regular spaces are pseudocomplete.
So I'd say that going via pseudocompleteness instead of Cech-completeness gives us a better common strengthening of Baire : it's a class of spaces that includes all completely metrisable spaces and all locally compact Hausdorff spaces, even all Cech-complete spaces, and pseudocompact completely regular spaces as well. All its members are Baire and it's closed under all products. And the Sorgenfrey line is pseudocomplete but not Cech-complete, as I explain in the comments.
But still, as you know, there is a product of two Baire spaces that is not Baire. But it does not happen for the nice and common classes of Baire spaces that happen to be pseudocomplete.