Holmstrom About Deligne cohomology

Levine mentioned the Thom-Sullivan cochain construction, in the MM book, or Bloch-Kriz. This gives a strictly multiplicative structure on the Godement resolution of a DGA, with rational coeffs. Can this be applied to the Deligne complex, and do we still need it at all???

Huber mentioned that something in LNM1604 could possible be used to get a strict product on Hodge/Deligne cohomology. Maybe this would happen by looking at the diagram given by inclusions of the $F^p$ and the constant things (with twists) into $\Omega^*$, and take some kind of homotopy (co?)limit of this.

A general question is still if we can consider Galois fixed points of a spectrum, in particular if we can relate Deligne cohom of real varieties to Deligne cohom of complex varieties, in the language of spectra.

nLab page on About Deligne cohomology