Place 1/2 the butter and 1/2 the garlic in a medium pot over medium-low heat. When the butter is melted, stir in the tomatoes, basil and lime juice. Just before the mixture comes to a boil, reduce the heat to low.
In a small pot, melt the remaining butter and mix in the remaining garlic.
Arrange the swordfish in a baking pan, and drizzle with the butter and garlic mixture from the small pot.
Bake the fish for 7 minutes in the preheated oven. Turn fish, and continue baking 7 minutes, or until easily flaked with a fork. Spoon the tomato mixture over the fish to serve. Also top with remaining butter and garlic sauce from the baking pan.