In our earlier paper “Continuous homology and real homotopy type”, we studied localization of simplicial spaces at the reals ans established an equivalence between the category of free nilpotent differential graded commutative algebras of finite type over the reals. In this paper, we extend these results by eliminating the nilpotent condition on the algebraic side, thus proving a conjecture of Sullivan. The main technical work consists in introducing local coefficients into continuous cohomology, the Serre spectral sequence and the constructions involved in real homotopy type.