We identify a certain class of persistence modules indexed over $\mathbb{R}^2$ that are decomposable into direct sums of indecomposable summands called blocks. The conditions on the modules are that they are both pointwise finite-dimensional (pfd) and exact. Our proof follows the same scheme as the one for pfd persistence modules indexed over $\mathbb{R}$, yet it departs from it at key stages due to the product order not being a total order on $\mathbb{R}^2$, which leaves some important gaps open. These gaps are filled in using more direct arguments. Our work is motivated primarily by the study of interlevel-sets persistence, although the proposed results reach beyond that setting.