Formality and splitting of real non-abelian mixed Hodge structures

We define and construct mixed Hodge structures on real schematic homotopy types of complex projective varieties, giving mixed Hodge structures on their homotopy groups and pro-algebraic fundamental groups. We also show that these split on tensoring with the ring R[x] equipped with the Hodge filtration given by powers of (x-i), giving new results even for simply connected varieties. Thus the mixed Hodge structures can be recovered from cohomology groups of local systems, together with the monodromy action at the Archimedean place. As the basepoint varies, these all become real variations of mixed Hodge structure.