Heat olive oil in a large skillet. Saute tomatoes, onions, and ginger in the hot oil until the tomatoes are just beginning to soften, 2 to 3 minutes. Stir collard green pieces into the tomato mixture; cook until the greens begin to soften, about 1 minute more.

Transfer tomato mixture to a slow cooker; add water, rice, and salt.

Cook on Medium until the water is completely absorbed, about 4 hours. Season with salt and freshly ground black pepper to taste.