  In this paper we introduce the notion of infinite dimensional Jacobi
structure to describe the geometrical structure of a class of nonlocal
Hamiltonian systems which appear naturally when applying reciprocal
transformations to Hamiltonian evolutionary PDEs. We prove that our class of
infinite dimensional Jacobi structures is invariant under reciprocal
transformations. The main technical tool is in a suitable generalization of the
classical Schouten-Nijenhuis bracket to the space of the so called quasi-local
multi-vectors, and a simple realization of this structure in the framework of
supermanifolds. These constructions are used to the computation of the
Lichnerowicz-Jacobi cohomologies of Jacobi structures. We also introduce the
notion of bi-Jacobi structures and consider the integrability of a system of
evolutionary PDEs that possesses a bi-Jacobi structure.
