Place oil in a medium-sized pan and warm over medium-high heat. Add sausage and cook until browned and crumbly, 5 to 7 minutes. Add onion and cook until translucent, 3 to 5 minutes. Add garlic and cook until slightly browned, about 2 minutes more. Transfer mixture to a slow cooker.

Add broth, water, and potatoes to the slow cooker. Season with oregano, basil, salt, and pepper. Cover and cook on High power for 4 to 6 hours.

Add kale to the soup in the slow cooker and let wilt slightly, about 5 minutes. Stir in coconut milk.