  Correlation functions involving products and ratios of half-integer powers of
characteristic polynomials of random matrices from the Gaussian Orthogonal
Ensemble (GOE) frequently arise in applications of Random Matrix Theory (RMT)
to physics of quantum chaotic systems, and beyond. We provide an explicit
evaluation of the large-$N$ limits of a few non-trivial objects of that sort
within a variant of the supersymmetry formalism, and via a related but
different method. As one of the applications we derive the distribution of an
off-diagonal entry $K_{ab}$ of the resolvent (or Wigner $K$-matrix) of GOE
matrices which, among other things, is of relevance for experiments on chaotic
wave scattering in electromagnetic resonators.
