  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 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. Unveiling the phase diagram of the non-interconverting
binary mixture gives unprecedented insight on the phase behaviors accessible to
the interconverting fluid, such as a liquid-liquid transition with a critical
point, or a singularity-free scenario, exhibiting thermodynamic anomalies
without polyamorphism. The model provides a unified theoretical framework to
describe supercooled water and a variety of polyamorphic liquids with
water-like anomalies.
