  The method of intersection spaces associates cell-complexes depending on a
perversity to certain types of stratified pseudomanifolds in such a way that
Poincar\'e duality holds between the ordinary rational cohomology groups of the
cell-complexes associated to complementary perversities. The cohomology of
these intersection spaces defines a cohomology theory HI for singular spaces,
which is not isomorphic to intersection cohomology IH. Mirror symmetry tends to
interchange IH and HI. The theory IH can be tied to type IIA string theory,
while HI can be tied to IIB theory. For pseudomanifolds with stratification
depth 1 and flat link bundles, the present paper provides a de Rham-theoretic
description of the theory HI by a complex of global smooth differential forms
on the top stratum. We prove that the wedge product of forms introduces a
perversity-internal cup product on HI, for every perversity. Flat link bundles
arise for example in foliated stratified spaces and in reductive Borel-Serre
compactifications of locally symmetric spaces. A precise topological definition
of the notion of a stratified foliation is given.
