It looks like I got my product construction a bit backwards -- the proper relation should be \$$(x, y) \le (x', y') \implies x \otimes y \le x' \otimes y'\$$. This should have been obvious to me before I posted, since tensor product and cartesian product are closely related (e.g. in the theory of [multilinear maps](https://en.wikipedia.org/wiki/Multilinear_map#Relation_to_tensor_products)).