We introduce the notion of duality solution for a two-phase compressible/incompressible fluid model, known as the hard-congestion model, on the real line, and additionally prove an existence result for this class of solutions. This system is derived from the analysis of a generalised Aw-Rascle system (a model for traffic flow and crowd dynamics). We prove that under suitable assumptions on the initial data, solutions to the Aw-Rascle system converge towards the so-called duality solutions, which have previously found applications in other systems which exhibit compressive dynamics. We also discuss uniqueness issues and prove that one can obtain weak solutions to the limiting system under stricter assumptions on the initial data.