We study the thermodynamic behavior of an Ising system on a Bethe lattice in which rearranging particles are "decorating" with their presence the bonds of the system, causing the local exchange coupling to depend on the decoration status. Such magnetic models have been proposed in efforts to understand the mechanisms responsible for the pairing of electrons in high-T, superconductivity. In order to study in some detail this aspect, we focus on the question of conditions under which particle pairing occurs, and more specifically, on the role of an external magnetic field. We find a low-temperature region of the phase diagram where significant particle clustering occurs when the field is introduced.