  In this paper, we study holomorphic rank-2 vector bundles on non-K\" ahler
elliptic surfaces. Our main tool for analysing these bundles is of course the
spectral cover. However, given the non-K\"{a}hler condition, the elliptic
surfaces we are considering do not have sections and gerbes naturally arise in
this context. The spectral construction presented in this paper is a
modification of the Fourier-Mukai transform for elliptic fibrations without a
section. After examining some of the properties of this Fourier-Mukai
transform, we give a complete classification of vector bundles on these
surfaces.
