Let $C_1$ and $C_2$ be two smooth, projective, and geometrically connected curves over a field $K$ of characteristic $0$. Assume there are two non-constant branch mapping of $K$-curves, $\phi_1\colon C_1\to P^1_K$ and $\phi_2\colon C_2\to P^1_K$ onto the projective line.
Taking function fields, one gets two finite extensions $F_1,F_2$ of $K(x)$ that are regular over $K$ (i.e. $K$ is algebraically closed in $F_1$ and in $F_2$).
Now assume that the compositum field $F_1F_2$ is also regular over $K$ and that $ F_1, F_2$ are linearly disjoint over $ K(x) $. This means that $C_1\times_{P^1} C_2$ is absolutely irreducible.
My question is whether in this setting it is also true that $C_1\times_{P^1} C_2$ is smooth, projective, and geometrically connected.
I do not think so. If you take $\phi_i:\mathbb{P}^1\to\mathbb{P}^1$ given by $\phi_i(x)=x^{i+1}$, then the fields are linearly disjoint, but the fiber product is singular.