  Given two locally conformal symplectic (LCS) structures on manifolds $M_1$
and $M_2$, we construct a natural $\R^+$-torsor of locally conformal symplectic
structures on a certain covering space $M_1 \boxplus M_2$ of $M_1 \times M_2$.
As the smooth construction of $M_1 \boxplus M_2$ is natural from the
perspective of flat line bundles, we use this language to phrase the LCS
theory. This construction shares many properties with, and in a sense
generalizes, the standard symplectic product. Notably, for a Hamiltonian
isotopy $\phi_t$ of an LCS manifold $M$, there is an associated Lagrangian
embedding $\Gamma(\phi_1) \colon M \hookrightarrow M \boxplus M$, in which
certain fixed points of $\phi_1$ are in bijection with intersection points of
$\Gamma(\phi_1)$ with the diagonal $\Delta = \Gamma(\mathrm{id})$. Using a
Lagrangian intersection of result of the first author and E. Murphy, we may
conclude that if $\phi_t$ is a $C^0$-small Hamiltonian isotopy, then the number
of fixed points of $\phi_1$ is bounded below by the rank of the Novikov theory
associated to the Lee class of the LCS structure on $M$. Finally, we end the
paper by constructing the suspension of a Lagrangian submanifold along a
Hamiltonian isotopy in the LCS theory, again generalizing the symplectic
setting.
