  We study 5-dimensional Riemannian manifolds that admit an almost contact
metric structure. In particular, we generalize the class of quasi-Sasaki
manifolds and characterize these structures by their intrinsic torsion. Among
other things, we see that these manifolds admit a unique metric connection that
is compatible with the underlying almost contact metric structure. Finally, we
construct a family of examples that are not quasi-Sasaki.
