For algebraic number fields $K$ with $s\>0$ real and $2t\>0$ complex embeddings and “admissible” subgroups $U$ of the multiplicative group of integer units of $K$ we construct and investigate certain $(s+t)$-dimensional compact complex manifolds $X(K,U)$. We show among other things that such manifolds are non-Kähler but admit locally conformally Kähler metrics when $t=1$. In particular we disprove a conjecture of I. Vaisman.