  Let M be an oriented compact 3-manifold and let T be a (loose) triangulation
of M, with ideal vertices at the components of the boundary of M and possibly
internal vertices. We show that any spin structure s on M can be encoded by
extra combinatorial structures on T. We then analyze how to change these extra
structures on T, and T itself, without changing s, thereby getting a
combinatorial realization, in the usual "objects/moves" sense, of the set of
all pairs (M,s). Our moves have a local nature, except one, that has a global
flavour but is explicitly described anyway. We also provide an alternative
approach where the global move is replaced by simultaneous local ones.
