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I am just posting my comment as an answer. For the field $\mathbb{R}$, the affine $\mathbb{R}$-scheme $\text{Spec}\ \mathbb{R}[x,y]/\langle x^2+y^2\rangle$ is integral and geometrically connected, but it is not geometrically irreducible. If $X_k$ is an integral, locally finite type $k$-scheme that is normal and geometrically connected, then $X_{\overline{k}}$ is irreducible. For geometric irreducibility, it is irreleven whether $k$ is perfect: the field extension $\overline{k}/k^{\text{sep}}$ is a universal homeomorphism. However, perfectness is relevant for reducedness.