Let V be a variety over a number field, and let K and L be two algebraically closed What is known about the points of $V(\bar K \otimes_{\bar Q} \bar L )$ ? Are there results claiming that points in $V(\bar K \otimes_{\bar Q} \bar L)$ reduce to points in $V(\bar K)$ and $V(\bar L)$, up to finite index etc ?

For example, if V is an Abelian variety, I think it is true that an infinitely divisible point in the group $V(\bar K \otimes_{\bar Q} \bar L)$ is a product of a point in $V(\bar K)$ and a point in $V(\bar L)$. (I do not have a proof; in any case, a good proof).

Are there similar results for other classes of varieties? What would such a statement look like?