You should treat tomatoes like fruit ('cuz that's what they are!).
Preheat the oven to 375°. In a blender, puree the tomato (you should have 1 cup puree). In a medium bowl, whisk the flour, baking powder, allspice and salt. In a large bowl, whisk the sugar, oil and tomato puree. Beat in the eggs, then the flour mixture. Mix in the pineapple, pecans, carrot and coconut. Divide the batter among 24 paper-lined muffin cups.
Bake until a toothpick comes out clean when inserted into the centers, about 18 minutes. Let cool.