  By a real alphabeta-geometry we mean a four-dimensional manifold M equipped
with a neutral metric h such that (M,h) admits both an integrable distribution
of alpha-planes and an integrable distribution of beta-planes. We obtain a
local characterization of the metric when at least one of the distributions is
parallel (i.e., is a Walker geometry) and the three-dimensional distribution
spanned by the alpha- and beta-distributions is integrable. The case when both
distributions are parallel, which has been called two-sided Walker geometry, is
obtained as a special case. We also consider real \alpha\beta-geometries for
which the corresponding spinors are both multiple Weyl principal spinors.
