  We develop the theory of subproduct systems over the monoid $\mathbb{N}\times
\mathbb{N}$, and the non-self-adjoint operator algebras associated with them.
These are double sequences of Hilbert spaces $\{X(m,n)\}_{m,n=0}^\infty$
equipped with a multiplication given by coisometries from $X(i,j)\otimes
X(k,l)$ to $X(i+k, j+l)$. We find that the character space of the norm-closed
algebra generated by left multiplication operators (the tensor algebra) is
homeomorphic to a Euclidean homogeneous algebraic variety intersected with a
unit ball. Certain conditions are isolated under which subproduct systems whose
tensor algebras are isomorphic must be isomorphic themselves. In the absence of
these conditions, we show that two numerical invariants must agree on such
subproduct systems. Additionally, we classify the subproduct systems over
$\mathbb{N}\times \mathbb{N}$ by means of ideals in algebras of non-commutative
polynomials.
