  Given a full right-Hilbert C*-module $\mathbf{X}$ over a C*-algebra $A$, the
set $\mathbb{K}_{A}(\mathbf{X})$ of $A$-compact operators on $\mathbf{X}$ is
the (up to isomorphism) unique C*-algebra that is strongly Morita equivalent to
the coefficient algebra $A$ via $\mathbf{X}$. As bimodule,
$\mathbb{K}_{A}(\mathbf{X})$ can also be thought of as the balanced tensor
product $\mathbf{X}\otimes_{A} \mathbf{X}^{\mathrm{op}}$, and so the latter
naturally becomes a C*-algebra. We generalize both of these facts to the world
of Fell bundles over groupoids: Suppose $\mathscr{B}$ is a Fell bundle over a
groupoid $\mathcal{H}$ and $\mathscr{M}$ an upper semi-continuous Banach bundle
over a principal right $\mathcal{H}$-space $X$. If $\mathscr{M}$ carries a
right-action of $\mathscr{B}$ and a sufficiently nice $\mathscr{B}$-valued
inner product, then its imprimitivity Fell bundle
$\mathbb{K}_{\mathscr{B}}(\mathscr{M})=\mathscr{M}\otimes_{\mathscr{B}}
\mathscr{M}^{\mathrm{op}}$ is a Fell bundle over the imprimitivity groupoid of
$X$, and it is the unique Fell bundle that is equivalent to $\mathscr{B}$ via
$\mathscr{M}$. We show that $\mathbb{K}_{\mathscr{B}}(\mathscr{M})$ generalizes
the 'higher order' compact operators of Abadie and Ferraro in the case of
saturated bundles over groups, and that the theorem recovers results such as
Kumjian's Stabilization trick.
