You can run power over coax! You can buy power-injecting splitters that were used to power old analog cameras. They basically just connect the cable to the 12V,

You can run power over coax! You can buy power-injecting splitters that were used to power old analog cameras. They basically just connect the cable to the 12V, sometimes directly but usually through some current-limiting safety switch. MoCA devices have a 100 Ohm internal resistor at the end to limit the cable echoes, so they are not affected by the DC on the cable.