Let $K$ be a non-algebraically closed field of characteristic $0$ and $X_K$ a smooth, projective, geometrically connected curve defined over $K$. If $F$ is a stable locally free sheaf on $X_K$, is it also necessarily stable on $X_\bar{K}$, where $\bar{K}$ denotes the algebraic closure of $K$.

I know that this is certainly not the case in characteristic $p$ but could not find an example where things go wrong in characteristic $0$. An example or reference would be very helpful.

Let $K$ be a non-algebraically closed field of characteristic $0$ and $X_K$ a smooth, projective, geometrically connected curve defined over $K$. If $F$ is a stable locally free sheaf on $X_K$, is it also necessarily stable on $X_\bar{K}$, where $\bar{K}$ denotes the algebraic closure of $K$?

I know that this is certainly not the case in characteristic $p$ but could not find an example where things go wrong in characteristic $0$. An example or reference would be very helpful.