The Euler Stratification for $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^n$

We study the Euler characteristic of a hypersurface in $(\mathbb{C}^*)^2 \times (\mathbb{C}^*)^n$ defined by a polynomial whose monomial support corresponds to lattice points in $\Delta_1 \times \Delta_1 \times \Delta_n$ as the coefficients of the defining polynomial vary. Each member of this hypersurface family corresponds to a three-way independence model from algebraic statistics, and the (signed) Euler characteristic is equal to the maximum likelihood degree (ML degree) of the model. We show in the case of $\Delta_1 \times \Delta_1 \times \Delta_1$ this Euler characteristic depends only on the vanishing patterns of the factors of the principal $A$-determinant, but this fails for $\Delta_1 \times \Delta_1 \times \Delta_n$ with $n \geq 2$. We prove that, for all $n\geq 1$, all positive integers up to the maximum possible ML degree can be realized as the Euler characteristic. Furthermore, we completely determine the Euler stratification for $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$ and provide partial information for $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^2$.