  A compact manifold $M$ together with a Riemannian metric $h$ on its universal
cover $\tilde M$ for which $\pi_1(M)$ acts by similarities is called a
similarity structure. In the case where $\pi_1(M) \not\subset
\mathrm{Isom}(\tilde M, h)$ and $(\tilde M, h)$ is reducible but not flat, this
is a Locally Conformally Product (LCP) structure. The so-called characteristic
group of these manifolds, which is a connected abelian Lie group, is the key to
understand how they are built. We focus in this paper on the case where this
group is simply connected, and give a description of the corresponding LCP
structures. It appears that they are quotients of trivial
$\mathbb{R}^p$-principal bundle over simply-connected manifolds by certain
discrete subgroups of automorphisms. We prove that, conversely, it is always
possible to endow such quotients with an LCP structure.
