Abstract : Keane, Berbee and others have studied the question of which specifications (aka $g$-functions) admit a unique Gibbs measure. Bramson and Kalikow constructed the first example of a regular and continuous specification which admits multiple measures. For every $p>2$, we construct a regular and continuous specification, whose variation is in $\ell^p$, that admits multiple Gibbs measures. This shows that a recent condition of Oberg and Johansson is tight. Joint work with Christopher Hoffman and Vladas Sidoravicius.