In a large 5-quart pot or Dutch oven, cook bacon over medium-low heat until browned and crisp, 8 to 10 minutes. Spoon off and discard all but 1 tablespoon fat. Add onion and carrots; cook, stirring occasionally, until softened, about 10 minutes. Add tomatoes and their juice (breaking tomatoes up with a spoon), clam juice, and 1 1/2 cups water; bring to a boil.
Add potatoes and thyme; reduce heat to simmer. Cook until potatoes are tender, but not falling apart, 15 to 20 minutes.
Add tilapia; cover, and cook until opaque and flaky, about 3 minutes. Season with salt and pepper. With a ladle, spoon solids and liquid into six soup bowls; serve immediately.
Tilapia fillets have one thick side and one thinner side. To cut tilapia, halve the fillets along the center line before cutting them into chunks.