  In this paper, we prove that a non-projective compact K\"ahler $3$-fold with
nef anti-canonical bundle is, up to a finite \'etale cover, one of the
following: a manifold with vanishing first Chern class; the product of a K3
surface and the projective line; or a projective space bundle over a
$2$-dimensional torus. This result extends Cao-H\"oring's structure theorem for
projective manifolds to compact K\"ahler manifolds in dimension $3$. For the
proof, we investigate the Minimal Model Program for compact K\"ahler $3$-folds
with nef anti-canonical bundles by using the positivity of direct image
sheaves, $\mathbb{Q}$-conic bundles, and orbifold vector bundles.
