Is it possible that A ∪ B ⊢ α ∧ ¬α, but there be no β such that A ⊢ β and B ⊢ ¬β?

Question

In other words, is it possible in classical logic that two sets of sentences together ($\mathbf{A} \cup \mathbf{B}$) imply a contradictory proposition ($\alpha \wedge \neg \alpha$), but they fail to individually imply a sentence that contradicts a sentence of the other set (e.g. $\mathbf{A} \vdash \neg\beta$ and $\mathbf{B} \vdash \beta$)?

I'd like to know whether this is possible or not, the proof justifying the answer, and the name of the theorem, if it has one.

I would be even more grateful if I'm referred to a textbook explaining this in detail.

Answers in the framework of non classical logics (intutionistic, paraconsistent, relevant, etc.) are also welcome.

Thanks in advance.