  A multisection is a decomposition of a manifold into 1-handlebodies, where
each subcollection of the pieces intersects along a 1-handlebody except the
global intersection which is a closed surface. These generalizations of
Heegaard splittings and Gay-Kirby trisections were introduced by Ben Aribi,
Courte, Golla and the author, who proved in particular that any 5-manifold
admits such a multisection. In arbitrary dimension, we show that two classes of
manifolds admit multisections: surface bundles and fiber bundles over the
circle whose fiber itself is multisected. We provide explicit constructions,
with examples.
