Mechanical chime issue only after installing 2nd Ring Pro

Hi! I have a mechanical chime with three doorbell hookup terminals (front, side, rear). I installed the first Ring Pro on the front, following the instructions perfectly. The little pro box was wires to the "front" terminal and the "transformer" terminal as instructed. It works perfectly. I then moved onto installing the second Ring Pro on the rear of the house. I installed the little pro box in the mechanical doorbell exactly as instructed, using the "rear" terminal and the "transformer" terminal. I then wired up the ring pro to the house, and began setting it up through the App exactly like I did the front door one. As soon as it connected to WiFi and registered, my mechanical chime began to constantly go off, without stopping. I removed the Ring pro and my mechanical chime stops. As soon as I hook back up the Ring Pro, the mechanical chime begins going off constantly, without stopping. What do I need to make this work correctly?!