Working in two batches, place in a blender tomatoes, tomato juice, ginger, coriander, sugar, oil, and 3 tablespoons lime juice; puree on high speed until liquefied. Pour batches into a large bowl. Season generously with salt and pepper; refrigerate until cold, about 1 hour. Before serving, taste for seasoning. If desired, thin with cold water.
In a small bowl, whisk yogurt and remaining tablespoon lime juice until smooth; season with salt and pepper. Ladle soup into four serving bowls; top each with a dollop of yogurt and, if desired, serve with toasted flatbread.