1. Heat olive oil in a large pot or Dutch oven over medium-high heat. Add shallots and garlic and cook, stirring, until garlic is golden, 3 to 4 minutes. Add tomatoes to pot and cook until almost dry, about 3 minutes. Peel 2 long strips of zest from orange (without white pith) and add to pot. Squeeze in orange's juice and stir in vermouth and saffron. Increase heat to high and bring mixture to a rapid boil.
2. Add mussels to pot, cover, and cook, shaking pot once or twice, until mussels begin to open, 3 to 4 minutes. Stir in basil and cook until all mussels open, about 2 more minutes. Divide mussels and broth among warm bowls (discard any unopened shells) and serve immediately.
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