  Let $W$ be a nonorientable $4$-dimensional handlebody without $3$- and
$4$-handles. We show that $W$ admits a Lefschetz fibration over the $2$-disk,
whose regular fiber is a nonorientable surface with nonempty boundary. This is
an analogue of a result of Harer obtained in the orientable case. As a
corollary, we obtain a $4$-dimensional proof of the fact that every
nonorientable closed $3$-manifold admits an open book decomposition, which was
first proved by Berstein and Edmonds using branched coverings. Moreover, the
monodromy of the open book we obtain for a given $3$-manifold belongs to the
twist subgroup of the mapping class group of the page. In particular, we
construct an explicit minimal open book for the connected sum of arbitrarily
many copies of the product of the circle with the real projective plane.
  We also obtain a relative trisection diagram for $W$, based on the
nonorientable Lefschetz fibration we construct, similar to the orientable case
first studied by Castro. As a corollary, we get trisection diagrams for some
closed $4$-manifolds, e.g. the product of the $2$-sphere with the real
projective plane, by doubling $W$. Moreover, if $X$ is a closed nonorientable
$4$-manifold which admits a Lefschetz fibration over the $2$-sphere, equipped
with a section of square $\pm 1$, then we construct a trisection diagram of
$X$, which is determined by the vanishing cycles of the Lefschetz fibration.
Finally, we include some simple observations about low-genus Lefschetz
fibrations on closed nonorientable $4$-manifolds.
