Let $\overline{\mathbf{Q}}\subset \mathbf{C}$ be the field of algebraic numbers.
Does there exist a number field $K$ with the following property?
There are embeddings $\sigma,\tau:K\to \overline{\mathbf{Q}}$, and a smooth projective geometrically connected curve $X$ over $K$, such that $X_{\overline{\mathbf{Q}},\sigma}$ is not isomorphic to $X_{\overline{\mathbf{Q}},\tau}$ in the category of curves over $\overline{\mathbf{Q}}$.
If yes, the genus of $X$ has to be positive.
Is there a difference between the above question and the following question?
Are there embeddings $\sigma,\tau:K\to \mathbf{C}$, and a smooth projective geometrically connected curve $X$ over $K$, such that $X_{\mathbf{C},\sigma}^{an}$ is not isomorphic to $X_{\mathbf{C},\tau}^{an}$ in the category of Riemann surfaces?