  A meromorphic quadratic differential on a punctured Riemann surface induces
horizontal and vertical measured foliations with pole-singularities. In a
neighborhood of a pole such a foliation comprises foliated strips and
half-planes, and its leaf-space determines a metric graph. We introduce the
notion of an asymptotic direction at each pole, and show that for a punctured
surface equipped with a choice of such asymptotic data, any compatible pair of
measured foliations uniquely determines a complex structure and a meromorphic
quadratic differential realizing that pair. This proves the analogue of a
theorem of Gardiner-Masur, for meromorphic quadratic differentials. We also
prove an analogue of the Hubbard-Masur theorem, namely, for a fixed punctured
Riemann surface there exists a meromorphic quadratic differential with any
prescribed horizontal foliation, and such a differential is unique provided we
prescribe the singular-flat geometry at the poles.
