  In this paper, we give an equivariant compactification of the space PFlat(S)
of homothety classes of half-translation structures on a compact, connected,
orientable surface S. We introduce the space PMix(S) of homothety classes of
mixed structures on S, that are CAT(0) tree-graded spaces in the sense of Drutu
and Sapir, with pieces which are R-trees and completions of surfaces endowed
with half-translation structures. Endowing Mix(S) with the equivariant Gromov
topology, and using asymptotic cone techniques, we prove that PMix(S) is an
equivariant compactification of PFlat(S), thus allowing us to understand in a
geometric way the degenerations of half-translation structures on S. We finally
compare our compactification to the one of Duchin-Leininger-Rafi, based on
geodesic currents on S, by the mean of the translation distances of the
elements of the covering group of S.
