  A locally conformally product (LCP) structure on compact manifold $M$ is a
conformal structure $c$ together with a closed, non-exact and non-flat Weyl
connection $D$ with reducible holonomy. Equivalently, an LCP structure on $M$
is defined by a reducible, non-flat, incomplete Riemannian metric $h_D$ on the
universal cover $\tilde M$ of $M$, with respect to which the fundamental group
$\pi_1(M)$ acts by similarities. It was recently proved by Kourganoff that in
this case $(\tilde M, h_D)$ is isometric to the Riemannian product of the flat
space $\mathbb{R}^q$ and an incomplete irreducible Riemannian manifold
$(N,g_N)$. In this paper we show that for every LCP manifold $(M,c,D)$, there
exists a metric $g\in c$ such that the Lee form of $D$ with respect to $g$
vanishes on vectors tangent to the distribution on $M$ defined by the flat
factor $\mathbb{R}^q$, and use this fact in order to construct new LCP
structures from a given one by taking products. We also establish links between
LCP manifolds and number field theory, and use them in order to construct large
classes of examples, containing all previously known examples of LCP manifolds
constructed by Matveev-Nikolayevsky, Kourganoff and Oeljeklaus-Toma.
