  Let $f : X \to S$ be a family of smooth projective algebraic varieties over a
smooth connected base $S$, with everything defined over
$\overline{\mathbb{Q}}$. Denote by $\mathbb{V} = R^{2i} f_{*} \mathbb{Z}(i)$
the associated integral variation of Hodge structure on the degree $2i$
cohomology. We consider the following question: when can a fibre
$\mathbb{V}_{s}$ above an algebraic point $s \in S(\overline{\mathbb{Q}})$ be
isomorphic to a transcendental fibre $\mathbb{V}_{s'}$ with $s' \in
S(\mathbb{C}) \setminus S(\overline{\mathbb{Q}})$? When $\mathbb{V}$ induces a
quasi-finite period map $\varphi : S \to \Gamma \backslash D$, conjectures in
Hodge theory predict that such isomorphisms cannot exist. We introduce new
differential-algebraic techniques to show this is true for all points $s \in
S(\overline{\mathbb{Q}})$ outside of an explicit proper closed algebraic subset
of $S$. As a corollary we establish the existence of a canonical
$\overline{\mathbb{Q}}$-algebraic model for normalizations of period images.
