  We revisit a long standing issue in the theory of disordered electron systems
and their effective description by a non-linear sigma model: the hyperbolic
Hubbard-Stratonovich (HS) transformation in the bosonic sector. For
time-reversal invariant systems without spin this sector is known to have a
non-compact orthogonal symmetry O(p,q). There exists an old proposal by
Pruisken and Schaefer how to do the HS transformation in an O(p,q) invariant
way. Giving a precise formulation of this proposal, we show that the HS
integral is a sign-alternating sum of integrals over disjoint domains.
