  We show that the birationality class of a quadric surface bundle over
$\mathbb{P}^2$ is determined by its associated cokernel sheaves. As an
application, we discuss stable-rationality of very general quadric bundles over
$\mathbb{P}^2$ with discriminant curves of fixed degree. In particular, we
construct explicit models of these bundles for some discriminant data. Among
others, we obtain various birational models of a nodal Gushel-Mukai fourfold,
as well as of a cubic fourfold containing a plane. Finally, we prove stable
irrationality of several types of quadric surface bundles.
