Oops, I missed this - sorry for posting the same answer! They must have gone up around the same time.

Sorry, I misread the question!

We have \begin{align*} X \times_{\mathrm{Spec}\ k} \mathrm{Spec}\ \overline{k} &= (X \times_{\mathrm{Spec}\ k'} \mathrm{Spec}\ k') \times_{\mathrm{Spec}\ k} \mathrm{Spec}\ \overline{k}\\ &= X \times_{\mathrm{Spec}\ k'} \mathrm{Spec}\ (k' \otimes_k \overline{k})\\ &= X \times_{\mathrm{Spec}\ k'} \left( \sqcup_{k' \hookrightarrow \overline{k}} \mathrm{Spec}\ \overline{k}\right) \\ &= \sqcup_{k' \hookrightarrow \overline{k}} \left(X \times_{\mathrm{Spec}\ k'} \overline{k}\right) \end{align*} This is not connected, since $k'/k$ is a non-trivial separable extension.

But an element of $K(X)$ which is algebraic over $k$ is certainly integral over $A$, so it is an element of $K(X)$. Let $k'$ be the separable closure of $k$ in $K(X)$; then we've shown that $X$ is naturally a $k'$-scheme ($k'$ doesn't depend on the choice of affine open), and that it is geometrically irreducible over $k'$ (since $k'$ is separably closed in $K(X)$).

I was surprised by the statement when $X$ is normal, and I couldn't find a proof online, so here's a sketch of the argument I worked out for future reference: We can work affine-locally on $X$, so assume $X = \mathrm{Spec}(A)$ for $A$ an integrally closed domain. It suffices to show that $A \otimes_k \overline{k}$ has a unique minimal prime ideal. Since field extensions are flat and $A$ injects into $K(X)$, it suffices to show that $K(X) \otimes_k \overline{k}$ has a unique minimal prime ideal. This is equivalent to $k$ being separably closed in $K(X)$.

Thanks, Jason! If you'll submit this as an answer, I'll accept it.

Thanks for the reference. The paper addresses the questions I had in mind, but for the sake of completeness, would you mind clarifying what it means to interpret a formula in the language of set theory in $V^{\mathrm{Sh}(\mathbb{P})}$, and what this has to do with the Kripke-Joyal semantics of the topos?

Which automorphic $L$-functions do not come from Galois representations? What is the explanation for them?