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 Abelien 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?