  We study framed translation surfaces corresponding to meromorphic
differentials on compact Riemann surfaces, for which a horizontal separatrix is
marked for each pole or zero. Such geometric structures naturally appear when
studying flat geometry surfaces "near" the Deligne-Mumford boundary.We compute
the number of connected components of the corresponding strata, and give a
simple topological invariant that distinguishes them. In particular we see that
for $g>0$, there are at most two such components, except in the hyperelliptic
case.
