Mix bananas, carrots, eggs, oil, honey, ginger and lemon juice, either by hand or with electric mixer on low speed.
Stir flour, cornmeal, slat, baking soda and allspice in a separate bowl.
Stir flour mixture and then walnuts into wet mixture, being careful not to overmix.
Pour into loaf pan and bake for 1 hour, or until a wooden toothpick inserted in center comes out clean.
Remove from pan and cool on a rack.
(You can grate ginger with a cheese grater and put it in the lemon juice while you prepare the rest of the mixture. Then strain the lemon juice and add it to the batter. That way you won't have pieces of ginger in the bread and the ginger flavor won't be too strong).