Preheat oven to 325F. Lightly grease a 9x13 baking pan and set aside.
In a large bowl, whisk eggs, milk, sugar, butter, coffee, vanilla, cinnamon, and salt until smooth. Add bread cubes to bowl and gently stir until bread is moistened.
Pour bread mixture into prepared pan and sprinkle chocolate chips over top. Bake bread pudding at 325F 50-55 minutes or until lightly browned (middle may still be a little jiggly---this is ok) Cool bread pudding slightly on a wire cooling rack.
Make the caramel sauce: Melt butter in a small saucepan over medium heat. Once melted, whisk in sugar, agave, and rum until smooth. Bring sauce to a boil over medium-heat until dark in color and slightly thickened. Remove from heat and whisk in salt, baking soda, and cream until smooth and pourable.
Serve bread pudding warm with warm caramel rum sauce. Enjoy!