I'm still trying to wrap my head around the "pathological" algebraic space $\mathbb{A}^1/\mathbb{Z}$; see Questions about the algebraic space $\mathbb{A}^1/\mathbb{Z}$ and Why is this not an algebraic space?
Let's put ourselves in a more general situation.
Let $X$ and $Y$ be finite type algebraic stacks over Spec $\mathbb{Z}$. Assume that $Y_{\mathbb{Q}}$ is isomorphic to $X_{\mathbb{Q}}$ over $\mathbb{Q}$.
Is there an etale open $U =$ Spec $O_{K,S}$ of Spec $\mathbb{Z}$ such that $X$ and $Y$ are isomorphic over $U$?
I think the answer is negative, but I can't see how to construct a counterexample. The usual spreading out results in the literature require "finite presentation", and $X$ is not necessarily of finite presentation over $\mathbb{Z}$ (nor $\mathbb{Q}$).
