Gallium based liquid metals (LM) have prospective biomedical, stretchable electronics, soft robotics, and energy storage applications, and are being widely adopted as thermal interface materials. The danger of gallium corroding most metals used in microelectronics requires the cumbersome addition of "barrier" layers or LM break-up into droplets within an inert matrix such as silicone oil. Such LM-in-oil emulsions are stabilized by native oxide on the droplets but have decreased thermal performance. Here we show that mixing of the silicone oil into an LM-air foam yields emulsions with inverted phases. We investigate the stability of these oil-in-LM emulsions through a range of processing times and oil viscosities, and characterize the impact of these parameters on the materials' structure and thermal property relationships. We demonstrate that the emulsion with 40 vol% of 10 cSt silicone oil provides a unique thermal management material with a 10 W m-1 K-1 thermal conductivity and an exterior lubricant thin film that completely prevents corrosion of contacting aluminum.