  A meromorphic differential on a Riemann surface is said to be {\it
real-normalized} if all its periods are real. Real-normalized differentials on
Riemann surfaces of given genus with prescribed orders of their poles form real
orbifolds whose topology is closely related to that of moduli spaces of Riemann
surfaces with marked points. Our goal is to develop tools to study this
topology. We propose a combinatorial model for the real normalized
differentials with a single order $2$ pole and use it to analyze the
corresponding absolute period foliation.
