  We study relatively semi-stable vector bundles and their moduli on
non-K\"ahler principal elliptic bundles over compact complex manifolds of
arbitrary dimension. The main technical tools used are the twisted
Fourier-Mukai transform and a spectral cover construction. For the important
example of such principal bundles, the numerical invariants of a 3-dimensional
non-K\"ahler elliptic principal bundle over a primary Kodaira surface are
computed.
