Heat the olive oil in a large heavy pot over medium heat. Add the onion, garlic, and bell pepper, and cook until softened and lightly browned, 8 to 10 minutes.
Stir in the paprika, salt, pepper, sugar, tomato paste, and tomato purée, and cook for 1 minute to develop the flavors. Add the diced tomatoes and cook for 5 minutes, stirring occasionally.
Pour in enough water to fill the pot three-quarters full. Bring to a boil, reduce to a simmer, cover, and cook for 20 minutes.
Blend the soup with an immersion blender until smooth. Add the rice and Swiss chard, return to a gentle simmer, and cook until the rice is tender, about 30 minutes.
Taste and adjust seasoning. Ladle into bowls and finish with a drizzle of olive oil and a pinch of dried chili. Serve hot.