  Motivated by recent experiments on liquid $^3$He reporting emergence of novel
superfluid phases in globally anisotropic aerogels, our previous theory on
superfluid $^3$He in globally anisotropic aerogels is extended to incorporate
effects of an anisotropy of the quasiparticle scattering cross section on the
strong coupling (SC) contributions to the Ginzburg-Landau (GL) free energy on
the basis of the spin-fluctuation (paramagnon) approach to the SC contributions
developed by Brinkman et al. [Phys. Rev. A {\bf 10}, 2386 (1974)]. In the
globally isotropic case, impurity effects on the SC correction destabilize the
A-phase even at higher pressures of about 30 (bar) and make the B-phase the
only state in equilibrium, while SC contributions accompanied by a global
stretched anisotropy to the GL quartic terms generally tend to broaden the
stability region of the A-phase compared with that of the B-phase. In
particular, in contrast to the cases in bulk and in the isotropic aerogel, the
SC corrections to the GL {\it quadratic} terms are not negligible in the
globally anisotropic case but may change the sign of the apparent anisotropy
depending on the magnitude of the frequency cutoff of the normal paramagnon
propagator. Based on this sign change of the apparent anisotropy, we discuss
different strange observations on superfluid $^3$He in porous media such as
disappearance of the polar superfluid phase at higher pressures seen in
nematically-ordered aerogels and absence of the B-phase and of the A-phase with
planar ${\hat {\bf l}}$-vector in a stretched aerogel.
