Let $M$ be a closed, connected, orientable 3-manifold with a Heegaard splitting $(F,H_1,H_2)$. Haken's Lemma states that if $M$ is reducible, i.e. if there is an embedded essential sphere in $M$, then there is an embedded essential sphere $S$ in $M$ such that $S\cap F$ is a simple closed curve, and hence $S\cap H_i$ are disks.

Is there a version of this lemma for essential tori embedded in $M$? In other words, can one say that if there is an embedded essential torus in $M$, then there is one which cuts each handlebody $H_i$ in an annulus?