  A meromorphic quadratic differential with poles of order two, on a compact
Riemann surface, induces a measured foliation on the surface, with a spiralling
structure at any pole that is determined by the complex residue of the
differential at the pole. We introduce the space of such measured foliations,
and prove that for a fixed Riemann surface, any such foliation is realized by a
quadratic differential with second order poles at marked points. Furthermore,
such a differential is uniquely determined if one prescribes complex residues
at the poles that are compatible with the transverse measures around them. This
generalizes a theorem of Hubbard and Masur concerning holomorphic quadratic
differentials on closed surfaces, as well as a theorem of Strebel for the case
when the foliation has only closed leaves. The proof involves taking a compact
exhaustion of the surface, and considering a sequence of equivariant harmonic
maps to real trees that do not have a uniform bound on total energy.
