We experimentally characterize the microstructure and rheology of a carefully designed mixture of immiscible fluids and near-neutral-wetting colloidal particles. Particle bridging across two fluid interfaces provides a route to highly stable gel-like emulsions at volume fractions of the dispersed phase well below the random close-packing limit for spheres. We investigate the microstructural origins of this behavior by confocal microscopy and reveal a percolating network of colloidal particles that serves as a cohesive scaffold, bridging together droplets of the dispersed phase. Remarkably, the mixture's salient rheological characteristics are governed predominantly by the solids loading and can be tailored irrespective of the droplet volume fraction. The identification of this rheological hallmark could provide a means toward the improved design of modern products that utilize solid-stabilized interfaces.