Have you tried using the Trim function from the Pathfinder panel/menu? I tried replicating your screenshot and got the results below:
You can also use the Divide function from the Pathfinder panel/menu to get the same results. Both of these actions will slice up your objects wherever they overlap, but you can fuse the shapes together again using the Merge function from the Pathfinder panel/menu.
North America
Europe, Middle East and Africa
Asia Pacific