If $X\_{\mathbb{Q}}$ is a curve over $\mathbb{Q}$, we get a curve $X\_{\mathbb{Q}\_p}$ over $\mathbb{Q}\_p$ for every prime $p$.

My question is about the reverse process. Say we are given curves $X\_{\mathbb{Q} \_ p}$ for every $p$ such that $X\_ { \bar {\mathbb{Q}} \_ p} \cong X\_ {\bar{ \mathbb{Q}} \_ q}$ for any two primes $p$ and $q$ (where the isomorphism is as schemes; alternatively, they are isomorphic when base-changed to an algebraically closed field that contains both $\mathbb{Q}\_p$ and $\mathbb{Q} \_ q$). Then is there a nice way to describe the obstruction for these models to come from a global curve $X\_{\mathbb{Q}}$ over $\mathbb{Q}$?