Is bimodality a sufficient condition for a first-order phase transition existence?

Here we present two explicit counterexamples to the widely spread beliefs about an exclusive role of bimodality as the first-order phase transition signal. On the basis of an exactly solvable statistical model generalizing the statistical multifragmentation model of nuclei, we demonstrate that the bimodal distributions can naturally appear both in infinite and in finite systems without a phase transition. In the first counterexample a bimodal distribution appears in an infinite system at the supercritical temperatures due to the negative values of the surface tension coefficient. In the second counterexample we explicitly demonstrate that a bimodal fragment distribution appears in a finite volume analog of a gaseous phase. In contrast to the statistical multifragmentation model, the developed statistical model corresponds to the compressible nuclear liquid with the tricritical endpoint located at one third of the normal nuclear density. The suggested parameterization of the liquid phase equation of state is consistent with the L. Van Hove axioms of statistical mechanics and it does not lead to an appearance of the nonmonotonic isotherms in the macroscopic mixed phase region which are typical for the classical models of the Van der Waals type. Peculiarly, such a way to account for the nuclear liquid compressibility automatically leads to an appearance of an additional state that in many respects resembles the physical antinuclear matter.