  Given a smooth prime Fano threefold $X$ of genus 7 we consider its
homologically projectively dual curve $\Gamma$ and the natural integral functor
$\Phi^{!}:D^b(X) \to D^b(\Gamma)$. We prove that, for $d\geq 6$, $\Phi^{!}$
gives a birational map from a component of the moduli scheme $M_X(2,1,d)$ of
rank 2 stable sheaves on $X$ with $c_1=1$, $c_2=d$ to a generically smooth
$2d-9$-dimensional component of the Brill-Noether variety
$W^{2d-11}_{d-5,5d-24}$ of stable vector bundles on $\Gamma$ of rank $d-5$ and
degree $5d-24$ with at least $2d-10$ sections. This map turns out to be an
isomorphism for $d=6$, and the moduli space $M_X(2,1,6)$ is fine. For general
$X$, this moduli space is a smooth irreducible threefold.
