In his ICM 2002 talk (Topology of singular algebraic varieties, available also on arXiv) B. Totaro says on p. 3 (of the arXiv version): "Using the method of Guillen and Navarro Aznar I was able to define the weight filtration for complex and real analytic spaces." However, no reference to this statement is given there. The definition is not explicitly given either, but from the preceding discussion a reasonable guess seems to be that one considers simply the Leray filtration of the open embedding into some compactification. The result will depend on the compactification, but if one compactification dominates another one, then the weight filtrations are the same (ibid, theorem 2.2).
I wasn't able to find a proof of the latter statement in the literature and I would like to ask if anyone knows a reference.