There are two metric Baire spaces, whose product is not Baire; a counterexample is given in the slides An absolute barely Baire space, L.F. Aurichi and G.A.A. Medina, 2020. We present Fleissner and Kunen's proof with applications of Clubs and stationary sets in $\omega_{1}$.
Fleissner, W. G.; Kunen, K., Barely Baire spaces, Fundam. Math. 101, 229-240 (1978). ZBL0413.54036.
