  The classical Mordell-Weil theorem implies that an abelian variety $A$ over a
number field $K$ has only finitely many $K$-rational torsion points. This
finitude of torsion still holds even over the cyclotomic extension $K^{\rm
cyc}=K\mathbb{Q}^{\mathrm{ab}}$ by a result of Ribet. In this article, we
consider the finiteness of torsion points of an abelian variety $A$ over the
infinite algebraic extension $K_B$ obtained by adjoining the coordinates of all
torsion points of an abelian variety $B$. Assuming the Mumford-Tate conjecture,
and up to a finite extension of the base field $K$, we give a necessary and
sufficient condition for the finiteness of $A(K_B)_{\rm tors}$ in terms of
Mumford--Tate groups. We give a complete answer when both abelian varieties
have dimension both three, or when both have complex multiplication.
