  The interplay between competing short-range (SR) and long-range (LR)
interactions can cause nontrivial structures in phase diagrams. Recently,
horn-shaped unusual structures were found by Monte Carlo simulations in the
phase diagram of the Ising antiferromagnet (IA) with infinite-range
ferromagnetic-like (F) interactions [Phys. Rev. B {\bf 93}, 064109 (2016); {\bf
96}, 174428 (2017)], and also in an IA with LR interactions of elastic origin
modeling spin-crossover materials [Phys. Rev. B {\bf 96}, 144425 (2017)]. To
clarify the nature of the phases associated with the horn structures, we study
the phase diagram of the IA model with infinite-range F interactions by
applying a variational free energy in a cluster mean-field (CMF) approximation.
While the simple Bragg-Williams mean-field theory for each sublattice does not
produce a horn structure, we find such structures with the CMF method. This
confirms that the local thermal fluctuations enabled by the multisite clusters
are essential for this phenomenon. We investigate in detail the structure of
metastable phases in the phase diagram. In contrast to the phase diagram
obtained by the Monte Carlo studies, we find a triple point, at which
ferromagnetic-like, antiferromagnetic-like, and disordered phases coexist, and
also six tristable regions accompanying the horn structure. We also point out
that several characteristic endpoints of first-order transitions appear in the
phase diagram. We propose three possible scenarios for the transitions related
to the tristable regions. Finally, we discuss the relation between the triple
point in this phase diagram and that of a possible lattice-gas model, in which
solid, liquid, and gas phases can coexist.
