  We study a compactification of the moduli space of theta characteristics,
giving a modular interpretation of the geometric points and describing the
boundary stratification. This space is different from the moduli space of spin
curves. The modular description and the boundary stratification of the new
compactification are encoded by a tropical moduli space. We show that this
tropical moduli space is a refinement of the moduli space of spin tropical
curves. We describe explicitly the induced decomposition of its cones.
