Elementary example of Baire spaces whose product is not Baire

It is known that there are Baire spaces $X$ and $Y$ whose product is not Baire, the simplest construction I know is due to Cohen and goes as follow:

Let $S$ be a stationary subset of $\omega_1$, then forcing with the poset $\Bbb P_S$ of countable subsets of $S$ closed in $\omega_1$ ordered by $p\leq q$ iff $q\subseteq p$ and $(p\setminus q)\cap(\cup q)=\varnothing$ adds a club contained in $S$ to $\omega_1$. The proof of this fact shows that $\Bbb P_S$ is ${<\omega_1}$-distributive (hence Baire in the topology whose open sets are the initial segments). Now let $S_1$ and $S_2$ be two disjoint clubs in $\omega_1$, then the poset $\Bbb P_{S_1}\times\Bbb P_{S_2}$ is not Baire otherwise forcing with it would add two disjoint clubs to $\omega_1$.

The whole proof uses a bunch of set theoretic facts:

• forcing with a ${<\kappa}$-distributive poset does not add sequences of length ${<\kappa}$ to the ground model.
• the filter $G\times H$ is $\Bbb P\times\Bbb Q$-generic over $M$ iff $G$ is $\Bbb P$-generic over $M$ and $H$ is $\Bbb Q$-generic over $M[G]$.
• If $\Bbb P$ and $\Bbb Q$ are separative posets then $\Bbb P\times\Bbb Q$ is Baire in $M$ iff $\Bbb P$ is Baire in $M$ and for every filter $G$ which is $\Bbb P$-generic over $M$, $\Bbb Q$ is Baire in $M[G]$.

There is another construction due to Kunen and Fleissner who, for every cardinal $\kappa$, constructed a family $\{X_\alpha\mid \alpha<\kappa\}$ of metrizable Baire spaces such that $\prod_{\alpha<\kappa}X_\alpha$ is nowhere Baire, meaning that it contains a countable family of dense open sets whose intersection is empty, while for every $\lambda<\kappa$, $\prod\{X_\alpha\mid\alpha\neq\lambda\}$ is Baire. This construction also uses stationary sets and set theoretic tools.

Suppose I want to show an example of two Baire spaces $X$ and $Y$ whose product is not Baire to a person who knows point-set topology but not a lot of set theory, is there a simple explicit construction of two spaces with that property? (Bonus points is the spaces are metrizable, or $X=Y$ or $X\times Y$ is nowhere Baire)

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