Bring water, broth, lentils, potatoes, tomato sauce, garlic, vinegar and basil to a boil. Reduce heat to medium-high. Cover and cook for 5 minutes.
Add broccoli, soup vegetables and black-eyed peas. Cover and cook for 5 minutes. Stir in rice. Cover and cook for 5 minutes. Remove from heat. Let stand for 5 minutes before serving.
Top each serving with a tablespoon of yogurt.