  A visible action on a complex manifold is a holomorphic action that admits a
$J$-transversal totally real submanifold $S$. It is said to be strongly visible
if there exists an orbit-preserving anti-holomorphic diffeomorphism $\sigma $
such that $\sigma |_S = \operatorname{id}_S$. Let $G$ be the Heisenberg group
and $H$ a non-trivial connected closed subgroup of $G$. We prove that any
complex homogeneous space $D = G^{\mathbb{C}}/H^{\mathbb{C}}$ admits a strongly
visible $L$-action, where $L$ stands for a connected closed subgroup of $G$
explicitly constructed through a co-exponential basis of $H$ in $G$. This leads
in turn that $G$ itself acts strongly visibly on $D$. The proof is carried out
by finding explicitly an orbit-preserving anti-holomorphic diffeomorphism and a
totally real submanifold $S$, for which the dimension depends upon the
dimensions of $G$ and $H$. As a direct application, our geometric results
provide a proof of various multiplicity-free theorems on continuous
representations on the space of holomorphic sections on $D$. Moreover, we also
generate as a consequence, a geometric criterion for a quasi-regular
representation of $G$ to be multiplicity-free.
