This paper is a continuation of parts I and II under the same title (Green & Naghdi 1995a, b, Proc. R. Soc. Lond. A 448, 335, 357). In contrast to the earlier two papers which dealt with single phase continua, this paper is concerned with a new thermomechanical theory of multiphase interacting continua. We use the same unified procedure as in part I but now the various energies that enter the balance of energy must be modified to accommodate energetic contributions arising from N interacting finite constituents. Again, our derivation of the various basic balance laws is effected by an appeal to the form invariance of the balance of energy and leads to a system of basic equations of mixtures which are automatically consistent with the balance of energy or the first law of thermodynamics for mixtures of N constituents.