We find a geometric invariant of isotopy classes of strongly irreducible Heegaard splittings of toroidal 3–manifolds. Combining this invariant with a theorem of R Weidmann, proved here in the appendix, we show that a closed, totally orientable Seifert fibered space has infinitely many isotopy classes of Heegaard splittings of the same genus if and only if has an irreducible, horizontal Heegaard splitting, has a base orbifold of positive genus, and is not a circle bundle. This characterizes precisely which Seifert fibered spaces satisfy the converse of Waldhausen’s conjecture.
"Non-isotopic Heegaard splittings of Seifert fibered spaces." Algebr. Geom. Topol. 6 (1) 351 - 372, 2006. https://doi.org/10.2140/agt.2006.6.351