To prepare pudding, combine first 4 ingredients in a large bowl; stir well with a whisk. Add sugar, rum, cinnamon, ginger, and nutmeg. Add bread to milk mixture in bowl; toss gently to coat. Spoon mixture into an 11 x 7-inch baking dish coated with cooking spray. Cover and chill 1 hour.
Preheat oven to 350°.
Place baking dish in a roasting pan; add hot water to pan to a depth of 1 inch. Cover and bake at 350° for 15 minutes. Uncover and bake an additional 15 minutes or until a knife inserted in center of pudding comes out clean. Let stand 15 minutes.
To prepare sauce, combine 1/4 cup water, juice, and guava paste in a small saucepan over low heat. Cook 10 minutes or until paste is melted, stirring frequently. Remove from heat; stir in the remaining 1/2 cup water. Serve with pudding.