  We show that the residual categories of quadric surface bundles are
equivalent to the (twisted) derived categories of some scheme under the
following hypotheses. Case 1: The quadric surface bundle has a smooth section.
Case 2: The total space of the quadric surface bundle is smooth and the base is
a smooth surface. We provide two proofs in Case 1 describing the scheme as the
hyperbolic reduction and as a subscheme of the relative Hilbert scheme of
lines, respectively. In Case 2, the twisted scheme is obtained by performing
birational transformations to the relative Hilbert scheme of lines. Finally, we
apply the results to certain complete intersections of quadrics.
