The addition of the polymer poly(diallyl dimethyl ammonium chloride) (polyDADMAC) to multilamellar vesicle dispersions of ditallowethylester dimethyl ammonium chloride (DDAC) causes the vesicle particles to aggregate and form a space-spanning network. These networks collapse under their own buoyant weight. The time evolution of the vesicle network height is investigated as a function of polymer concentration, vesicle volume fraction, and initial sample height. A poroelastic consolidation model accurately relates the time scales of the network collapse to the balance of gravitational, viscous, and elastic forces. Network formation is consistent with an arrested spinodal decomposition of the colloid-polymer mixture; however, while such networks usually lead to nonequilibrium gels, the buoyant stresses are sufficient to drive the phase separation of this colloid-polymer mixture to completion.