Something more basic is true. Strongly irreducible Heegaard splittings act a lot like incompressible surfaces (of which fibers are a special case). Here are two results as evidence.
First, suppose that the three-manifold is equipped with a triangulation. Then Haken showed that incompressible surfaces can be normalized. That is, isotoped to intersect every tetrahedron in a collection of standard disks; normal triangles and normal quads. Rubinstein (and Stocking) showed that strongly irreducible splittings can be "almost normalized"; in every tetrahedron it has normal disks and in at most one tetrahedron there is an almost normal annulus or octagon. Michelle Stocking's thesis is a standard reference for this material. Many people (Rubinstein, Hass, Bachman, Scott, ...) will say in public that the above is a PL version of an analytic truth: incompressible surfaces can be isotoped to be minimal surfaces of index zero while strongly irreducible surfaces can be made minimal of index one.
Second, suppose that $S, T$ are surfaces, with $T$ incompressible. If $S$ is also incompressible then it is an exercise in innermost disks to show that, after isotoping $S$ to meet $T$ minimally, all curves of intersection are essential on both surfaces. If $S$ is instead strongly irreducible then a one-parameter sweep-out argument, followed by an innermost disk argument shows that there is some position of $S$ where all curves of intersection are essential on both surfaces.
Ok, third (I couldn't resist), suppose that the ambient three-manifold is hyperbolic. Minsky's approach to the ending lamination conjecture says that the geometry "around" the incompressible surface or a strongly irreducible splitting can be modelled using "blocks" based on the four-holed sphere or once-holed torus, where the blocks are glued to each other vertically using pairs of pants and horizontally using solid tori (Margulis tubes).
