Liquid polyamorphism is the intriguing possibility for a single component substance to exist in multiple liquid phases. We propose a minimal model for this phenomenon. Starting with a classical binary lattice model with critical azeotropy and liquid-liquid demixing, we allow interconversion of the two species, turning the system into a single-component fluid with two states differing in energy and entropy. Changing one interaction parameter allows to continuously switch from a liquid-liquid transition, terminated by a critical point, to a singularity-free scenario, exhibiting water-like anomalies but without polyamorphism. This resolves a controversy about how a liquid-liquid critical point can be found or not in simulations. The model provides a unified theoretical framework to describe supercooled water and a variety of polyamorphic liquids with water-like anomalies.
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