  We associate to each algebraic variety defined over $\mathbb{R}$ a filtered
cochain complex, which computes the cohomology with compact supports and
$\mathbb{Z}\_2$-coefficients of the set of its real points. This filtered
complex is additive for closed inclusions and acyclic for resolution of
singularities, and is unique up to filtered quasi-isomorphism. It is
represented by the dual filtration of the geometric filtration on semialgebraic
chains with closed supports defined by McCrory and Parusi\'nski, and leads to a
spectral sequence which computes the weight filtration on cohomology with
compact supports. This spectral sequence is a natural invariant which contains
the additive virtual Betti numbers. We then show that the cross product of two
varieties allows us to compare the product of their respective weight complexes
and spectral sequences with those of their product, and prove that the cup and
cap products in cohomology and homology are filtered with respect to the real
weight filtrations.
