  We study the asymptotic non-vanishing of syzygies for products of projective
spaces. Generalizing the monomial methods of Ein, Erman, and Lazarsfeld
\cite{einErmanLazarsfeld16} we give an explicit range in which the graded Betti
numbers of $\mathbb{P}^{n_1}\times \mathbb{P}^{n_2}$ embedded by
$\mathcal{O}_{\mathbb{P}^{n_1}\times\mathbb{P}^{n_2}}(d_1,d_2)$ are non-zero.
These bounds provide the first example of how the asymptotic syzygies of a
smooth projective variety whose embedding line bundle grows in a semi-ample
fashion behave in nuanced and previously unseen ways.
