  The paper computes the Witt-sheaf cohomology rings of partial flag varieties
in type A in terms of the Pontryagin classes of the subquotient bundles. The
proof is based on a Leray-Hirsch-type theorem for Witt-sheaf cohomology for the
maximal rank cases, and a detailed study of cohomology ring presentations and
annihilators of characteristic classes for the general case. The computations
have consequences for the topology of real flag manifolds: we show that all
torsion in the integral cohomology is 2-torsion, which was not known in full
generality previously. This allows for example to compute the Poincar\'e
polynomials of complete flag varieties for cohomology with twisted integer
coefficients. The computations also allow to describe the Chow-Witt rings of
flag varieties, and we sketch an enumerative application to counting flags
satisfying multiple incidence conditions to given hypersurfaces.
