  We extend our recent study of winding number density statistics in Gaussian
random matrix ensembles of the chiral unitary (AIII) and chiral symplectic
(CII) classes. Here, we consider the chiral orthogonal (BDI) case which is the
mathematically most demanding one. The key observation is that we can map the
topological problem on a spectral one, rendering the toolbox of random matrix
theory applicable. In particular, we employ a technique that exploits
supersymmetry structures without reformulating the problem in superspace.
