Crush tomatoes and juices in a large bowl with clean hands into small pieces. Heat oil in a soup pot over medium.
Cook onion 8 minutes until very soft; add garlic for 30 seconds. Add tomatoes with juices, broth, salt, and pepper.
Simmer uncovered 22 to 25 minutes, stirring occasionally and crushing large pieces against pot wall with a spoon. Do not blend.
Turn off heat; wait until bubbling stops. Whisk mascarpone with 1/2 cup hot soup until smooth; stir into pot.
Fold in basil; divide among 4 bowls, retaining soft tomato pieces in broth.