A specification on a shift space is a collection of conditional measures that describe the probability of seeing a particular finite configuration conditioned on the complement sigma-algebra. A particularly interesting class of specifications are the Gibbsian ones, which can be defined through a "nice" set of interactions on the space of configurations. Two famous theorems of Kozlov and Sullivan give partial answers to the question of when a continuous specification on a full shift is in fact Gibbsian: Kozlov's theorem states that every continuous specification is Gibbs by a nice interaction, but this interaction is not necessarily shift-invariant, while Sullivan shows that every continuous specification is Gibbs by a "not so nice" interaction which is shift invariant. The question of whether the non-shift invariance in Kozlov's proof is a fundamental part of it remained an "annoying" problem up to now. We provide a solution to this "annoying" problem. We show that there exist continuous specifications that can not be realized by a "nice" and shift-invariant interaction. This is work in collaboration with Ricardo Gómez-Aíza, Brian Marcus, Tom Meyerovitch and Siamak Taati.