  We combine experiments, theory, and simulations to investigate the
coexistence of nonequilibrium phases emerging from interacting colloidal
particles that are electrokinetically propelled in a nematic liquid crystal
solvent. We directly determine the mechanical pressure within the radial
assemblies and measure a non-equilibrium equation of state for this athermal
driven system. A generic model combines phoretic propulsion with the interplay
between electrostatic effects and liquid-crystal-mediated hydrodynamics, which
are effectively cast into a long-range interparticle repulsion, while
elasticity plays a subdominant role. Simulations based on this model explain
the observed collective organization process and phase coexistence
quantitatively. Our colloidal assemblies provide an experimental test-bed to
investigate the fundamental role of phoretic pressure in the organization of
driven out-of-equilibrium matter.
