Haken's Lemma for Essential Tori

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?

Yes. If the splitting surface $F$ is strongly irreducible then, after an isotopy you can assume that $T$ meets each of $H_1$ and $H_2$ in collections of disjoint incompressible annuli. This has been independently proved by Kobayashi, Thompson, and Hempel. For references, please see the bibliography of Hempel's paper "3-manifolds as viewed from the curve complex".
• Thanks @Sam. Why did you edit your post as "$T$ meets each of $H_1$ and $H_2$ in collections of disjoint incompressible annuli"? I just checked Hempel's paper, and he proves that each intersection is an essential annulus in Lemma 3.6. – Mustafa Nov 2 '16 at 0:56