  We define inverse semigroup actions on topological groupoids by partial
equivalences. From such actions, we construct saturated Fell bundles over
inverse semigroups and non-Hausdorff \'etale groupoids. We interpret these as
actions on C*-algebras by Hilbert bimodules and describe the section algebras
of these Fell bundles.
  Our constructions give saturated Fell bundles over non-Hausdorff \'etale
groupoids that model actions on locally Hausdorff spaces. We show that these
Fell bundles are usually not Morita equivalent to an action by automorphisms.
That is, the Packer-Raeburn Stabilisation Trick does not generalise to
non-Hausdorff groupoids.
