  We construct a quantisation of the Teichmueller spaces of super Riemann
surfaces using coordinates associated to ideal triangulations of super Riemann
surfaces. A new feature is the non-trivial dependence on the choice of a spin
structure which can be encoded combinatorially in a certain refinement of the
ideal triangulation. By constructing a projective unitary representation of the
groupoid of changes of refined ideal triangulations we demonstrate that the
dependence of the resulting quantum theory on the choice of a triangulation is
inessential.
