Preheat the oven to 350 degrees F (175 degrees C).

Melt 1/4 cup butter with 1 teaspoon garlic in a medium pot over medium-low heat. Stir in tomatoes, basil, and lime juice. Just before mixture comes to a boil, reduce heat to low.

Melt remaining butter in a small pot; mix in remaining garlic.

Arrange swordfish in a baking pan; drizzle with garlic butter.

Bake fish in the preheated oven for 7 minutes. Flip fish; continue baking until fish easily flakes with a fork, about 7 more minutes.

Spoon tomato mixture over fish to serve. Top with pan juices.