  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.
