We investigate micrometer-sized flat droplets consisting of an isotropic core surrounded by a nematic rim in freely suspended smectic A liquid-crystal films. In contrast to purely isotropic droplets which are characterized by a sharp edge and no long-range interactions, the nematic fringe introduces a continuous film thickness change resulting in long-range mutual attraction of droplets. The coalescence scenario is divided in two phases. The first one consists in the fusion of the nematic regions. The second phase involves the dissolution of a thin nematic film between the two isotropic cores. The latter has many similarities with the rupture of thin liquid films between droplets coalescing in an immiscible viscous liquid.