  In this paper we introduce the concept of $\mathit{integral}$
$\mathit{Frobenius}$ to formulate an integral analogue of the classical
compatibility condition linking the collection of rational Tate modules
$V_\lambda(A)$ arising from abelian varieties over number fields with real
multiplication. Our main result gives a recipe for constructing an integral
Frobenius when the real multiplication field has class number one. By
exploiting algorithms already existing in the literature, we investigate this
construction for three modular abelian surfaces over $\mathbf{Q}$.
