  We restore part of the thermodynamic formalism for some renormalized measures
that are known to be non-Gibbsian. We first point out that a recent theory due
to Pfister implies that for block-transformed measures free energies and
relative entropy densities exist and are conjugate convex functionals. We then
determine a necessary and sufficient condition for consistency with a
specification that is quasilocal in a fixed direction. As corollaries we obtain
consistency results for models with FKG monotonicity and for models with
appropriate "continuity rates". For (noisy) decimations or projections of the
Ising model, these results imply almost quasilocality of the decimated "+" and
"-" measures.
