  Over the past few decades, numerous $N$-phase incompressible
diffuse-interface flow models with non-matching densities have been proposed.
Despite aiming to describe the same physics, these models are generally
distinct, and an overarching modeling framework is absent. This paper provides
a unified framework for $N$-phase incompressible Navier-Stokes Cahn-Hilliard
Allen-Cahn mixture models with a single momentum equation. The framework
naturally emerges from continuum mixture theory, exhibits an energy-dissipative
structure, and is invariant to the choice of fundamental variables. This opens
the door to exploring connections between existing $N$-phase models and
facilitates the computation of $N$-phase flow models rooted in continuum
mixture theory.
