I am interested in knowing examples of when $\mathcal{B}(X\times Y) \neq \mathcal{B}(X) \times\mathcal{B}(Y)$. By allowing $|X|,|Y|$ to be large we can provide a trivial counterexample, as in the one given at https://mathoverflow.net/questions/39882/product-of-borel-sigma-algebras. I know that if $X,Y$ are separable metrizable then the the product equality holds. Do we have any counterexamples where $|X|,|Y| \leq |\mathbb{R}|$?
In particular I am interested in knowing whether $X=Y=C(\mathbb{R})$ (equipped with the subspace topology inherited from $\mathbb{R}^{\mathbb{R}}$) provides the necessary counterexample.