Heat oil in a large pot over medium-high heat and add sausage. Cook, stirring often, until sausage is browned and fat has rendered out, about 5 minutes.
Add onion and peppers and cook, stirring constantly until onion softens, about 4 minutes. Add potatoes and cook, stirring often, 2 minutes. Add turnip greens, Hoppin' John, diced tomatoes and green chiles, navy beans, water, and bouillon paste; stir to combine.
Bring mixture to a boil, cover, reduce heat and simmer until potatoes are tender, 25 to 30 minutes.
