Neutral hydrogen around high-z QSO and an optical depth tau ~ 0.17 can be reconciled if reionization is more complex than a single transition at z ~ 6-8. Tracing its details could shed a new light on the first sources of radiation. Here we discuss how far such details can be inspected through planned experiments on CMB large-scale anisotropy and polarization, by simulating an actual data analysis. By considering a set of double reionization histories of Cen (2003) type, a relevant class of models not yet considered by previous works, we confirm that large angle experiments rival high resolution ones in reconstructing the reionization history. We also confirm that reionization histories, studied with the prior of a single and sharp reionization, yield a biased tau, showing that this bias is generic. We further find a monotonic trend in the bias for the models that we consider, and propose an explanation of the trend, as well as the overall bias. We also show that in long-lived experiments such a trend can be used to discriminate between single and double reionization patterns.
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