  We use rewriting systems to spell out cup-products in the (twisted)
cohomology groups of a product of surface groups. This allows us to detect a
non-trivial obstruction bounding from below the effective topological
complexity of an orientable surface with respect to its antipodal involution.
Our estimates are at most one unit from being optimal, and are closely related
to the (regular) topological complexity of non-orientable surfaces.
