We call a topological space $(X,\tau)$ *product-decomposable* if there is an index set $I$ and subsets $X_i\subseteq X$ for $i\in I$ such that $|X_i| > 1$ and $X \cong \prod_{i\in I} X_i$ where each $X_i$ is endowed with the subspace topology.

If a space $(X,\tau)$ is not product-decomposable and $|X|>1$, we call it *prime*. (Is there established terminology for this?)

If $X_1, X_2$ are prime spaces, is it possible that there are prime topological spaces $Y_1, Y_2$ such that $X_i \not\cong Y_j$ for $i,j\in\{1,2\}$ but $(X_1\times X_2) \cong (Y_1\times Y_2)$?