We present a unified approach to establishing the Gibbsian character of a wide class of non-Gibbsian states, arising in the Renormalisation Group theory. Inside the realm of the Pirogov-Sinai theory for lattice spin systems, we prove that RG transformations applied to low temperature phases give rise to weakly Gibbsian measures. In other words, we show that the Griffiths-Pearce-Israel scenario of RG pathologies is carried by atypical configurations. The renormalized measures are described by an effective interaction, with relative energies well-defined on a full measure set of configurations. In this way we complete the first part of the Dobrushin Restoration Program: to give a Gibbsian description to non-Gibbsian states. A disagreement percolation estimate is used in the proof to bound the decay of quenched correlations through which the interaction potential is constructed. The percolation is controlled via a novel type of pathwise large deviation theory.