Say we have a model $M$ of a theory $T$ of some core fuzzy logic.

When dealing with compactness, we run in to a situation where the new model being built (by the use of compactness over $M$), will necessarily need a newer BL-algebra for the truth in it to be evaluated (for example, see here: Compactness and completeness in Gödel logic).

It is known that certain logics do not have this problem (the easiest being standard first order logic).

My question is, is there a classification on BL-algebras that says the BL-algebras having this property will allow for compactness (in the classical sense) to go through?

Edit: Also I know that there are studies of this question from the viewpoint; Which logics have (classical) compactness. I would be very thankful if someone could point me in the direction of a comprehensive survey of these results.