  Polyhedral K\"ahler surfaces are a class of complex surfaces, which are flat
everywhere except on a two-dimensional skeleton. They are defined as a
generalisation of the "gluing a polygon side by side" construction of flat
Riemann surfaces.
  In this article, we introduce two classes of polyhedral K\"ahler surfaces
with trivial holonomy: products of polyhedral K\"ahler curves with zero
holonomy and ramified coverings of tori, and prove that none of these classes
is contained in the other. Existence of other types of polyhedral K\"ahler
surfaces is still open.
