Pour tomatoes and juices into a large bowl and crush the tomatoes into small pieces with clean hands. Heat olive oil in a soup pot over medium heat.
Cook onion for 8 minutes until very soft, then stir in garlic for 30 seconds. Add crushed tomatoes, their juices, broth, salt and pepper.
Simmer uncovered for 22 to 25 minutes, stirring occasionally and pressing any large tomato pieces against the side with a spoon; do not blend.
Turn off the heat and wait until bubbling stops. Whisk mascarpone in a bowl with 1/2 cup hot soup until smooth, then stir that mixture back into the pot.
Fold in sliced basil and divide among 4 bowls, keeping the soft tomato pieces in the creamy broth.