  A locally conformally symplectic (LCS) form is an almost symplectic form
$\omega$ such that a closed one-form $\theta$ exists with
$d\omega=\theta\wedge\omega$. A fiber bundle with LCS fiber $(F,
\omega,\theta)$ is called LCS if the transition maps are diffeomorphisms of $F$
preserving $\omega$ (and hence $\theta$). In this paper, we find conditions for
the total space of an LCS fiber bundle to admit an LCS form which restricts to
the LCS form of the fibers. This is done by using the coupling form introduced
by Sternberg and Weinstein, \cite{gls}, in the symplectic case. The
construction is related to an adapted Hamiltonian action called twisted
Hamiltonian which we study in detail. Moreover, we give examples of such
actions and discuss compatibility properties with respect to LCS reduction of
LCS fiber bundles. We end with a glimpse towards the locally conformally
K\"ahler case.
