  Let $M\stackrel\pi \arrow X$ be a principal elliptic fibration over a Kaehler
base $X$. We assume that the Kaehler form on $X$ is lifted to an exact form on
$M$ (such fibrations are called positive). Examples of these are regular
Vaisman manifolds (in particular, the regular Hopf manifolds) and
Calabi-Eckmann manifolds. Assume that $\dim M > 2$. Using the Kobayashi-Hitchin
correspondence, we prove that all stable bundles on $M$ are flat on the fibers
of the elliptic fibration. This is used to show that all stable vector bundles
on $M$ take form $L\otimes \pi^* B_0$, where $B_0$ is a stable bundle on $X$,
and $L$ a holomorphic line bundle. For $X$ algebraic this implies that all
holomorphic bundles on $M$ are filtrable (that is, obtained by successive
extensions of rank-1 sheaves). We also show that all positive-dimensional
compact subvarieties of $M$ are pullbacks of complex subvarieties on $X$.
