Prepare the navel oranges, blood orange, grapefruit, and tangerines following the method in the photos. Put all the citrus sections in a bowl; squeeze in the juices from the leftover membranes. Cover and chill.
Slice the ends of the citrus to give you a stable base. Set the fruit on one end.
Slice off the peel, following the fruit's contours. Be sure to remove the white pith.
Slice along the membranes to free the sections; try to keep the sections intact.
Slice the top and bottom from the pineapple and rest the fruit on one of its flat cut surfaces. Cut off the pineapple skin, using the tip of a vegetable peeler to remove the eyes. Quarter it lengthwise, slice the core from each quarter, and cut each quarter into 1/2-inch pieces. Cover and chill.
A half hour before serving, combine the brown sugar, sparkling wine, and rum in a large bowl. Add the citrus and pineapple. Peel and slice the banana into 1/2-inch pieces; add it to the salad. Prepare the apple and the pear as follows: peel, quarter, core, and slice each quarter into four pieces, and add to the salad. Toss gently to combine thoroughly. Let sit covered at room temperature for about 30 minutes. Serve in small bowls with cookies on the side.