  The well-known Hammersley-Clifford theorem states (under certain conditions)
that any Markov random field is a Gibbs state for a nearest neighbor
interaction. In this paper we study Markov random fields for which the proof of
the Hammersley-Clifford theorem does not apply. Following Petersen and Schmidt
we utilize the formalism of cocycles for the homoclinic equivalence relation
and introduce "Markov cocycles", reparametrisations of Markov specifications.
The main part of this paper exploits this to deduce the conclusion of the
Hammersley-Clifford theorem for a family of Markov fields which are outside the
theorem's purview where the underlying graph is $\mathbb{Z}^d$. This family
includes all Markov random fields whose support is the d-dimensional "3-colored
chessboard". On the other extreme, we construct a family of shift-invariant
Markov random fields which are not given by any finite range shift-invariant
interaction.
