For countable-to-one transitive Markov maps, we show that the natural extensions of invariant ergodic weak Gibbs measures absolutely continuous with respect to weak Gibbs conformal measures possess a version of the u-Gibbs property. In particular, if dynamical potentials admit generalized indifferent periodic points then the natural extensions exhibit a non-Gibbsian character in statistical mechanics. Our results can be applicable to certain non-hyperbolic number-theoretical transformations of which natural extensions possess unstable (respectively stable) leaves with subexponential expansion (respectively contraction).