  We study the existence of realistic heterotic vacua on a new Abelian surface
fibered Calabi-Yau threefold X with Z_8 x Z_8 fundamental group. Our main
result is a no-go theorem, which says that (under mild assumptions) there is no
stable holomorphic vector bundle on X satisfying the constraints required by
global consistency of the heterotic vacuum and phenomenology. To prove the
theorem we explore in some detail the Fourier-Mukai transform of vector bundles
on Abelian surface fibrations.
