Preheat the oven to 325 degrees F (165 degrees C). Grease a 9x13-inch baking dish.

Chop peaches and place in a large bowl; mash peaches lightly using a fork. Combine condensed milk and beaten eggs in a separate bowl; add to peaches and mix well. Stir in hot water, melted butter, cinnamon, and vanilla. Fold in bread until bread is completely moistened. Transfer mixture to the prepared baking dish.

Bake in the preheated oven until a knife inserted into the center of pudding comes out clean, about 1 hour and 10 minutes.

Meanwhile, make sauce: Combine brown sugar, butter, corn syrup, and rum in a small saucepan. Bring to a boil over medium heat and simmer until just slightly thickened, 3 to 4 minutes. Remove from heat and cover to keep warm.

Remove pudding from the oven and allow to cool for about 10 minutes before serving. Serve warm with caramel sauce. Cool any leftover pudding completely and store, covered, in the refrigerator.