For the turkey: The day before serving, heat 3 quarts of the water with salt and sugar in a large pan, stirring to dissolve salt and sugar. Remove from heat, add remaining 1 quart cold water, and stir. Set brine aside to cool to room temperature.
Put turkey in a large container and cover with the brine. Cover and refrigerate for 4 to 5 hours. (If you want to brine the turkey overnight, use half the amount of salt and sugar.)
For rub: Crumble annatto paste into a food processor and blend with the garlic, oregano, coriander, black pepper, salt, and allspice. Add the fruit juices and the olive oil and process to make a pasty sauce. Drain and pat the turkey very dry. Smear the rub all over the bird. Cover and refrigerate overnight.
Prepare an outdoor grill with a large medium-heat fire for direct and indirect grilling. Position a drip pan under the grate on indirect side of the grill.
Place turkey, breast side up, over the drip pan and grill, covered, until meat is cooked about halfway through, about 50 minutes. Turn and cook until an instant-read thermometer inserted in the thigh registers 170 degrees F, about 50 minutes more. Move turkey to direct heat and rotate to evenly brown the skin in the last 10 minutes of cooking. Transfer to a carving board, cover, and let rest 10 minutes before carving.
Shopsmart: Annatto paste is a Mexican flavoring composed of ground annatto, or achiote, seeds mixed with herbs-usually oregano-and spices. The rust-color paste is then pressed into a compact brick. Look for it in the Hispanic section of your grocery.
Cook's Note: The turkey can be grilled as soon as it's rubbed with the spice paste, and it will be great. But if you have time, let it marinate-it will be awesome.
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