The easiest way to diagnose the problem is with the correct wiring diagram. When you actuate the left side and both rear lights are flashing that indicates power from the flasher unit to both bulbs instead of just the single bulb. There are two wires that go to the bulbs, one is power and the other is ground. The lights light when power goes through the bulb to ground. The flasher unit turns the power on and off using a timer based on resistance, that's all it does. To get both rear lights on means power has to be applied to both bulbs at the same time, logically that can only happen if they share a common power source. What year is your Speedmaster? If I can look up the wiring diagram I might be able to help you sort the problem. One other thought is to return the system back to origoinal and see if the problem is still there.
Older bikes have a flasher unit independent of the ECU, newer bikes have the flasher as part of the ECU. My 2022 Speed Triple RS 1200 has it's flasher in the ECU and is progammable (mY 98 Trophy uses an independent flasher). The only way that could power both bulbs in the rear or front is if the wires were cross connected. And I do have a hazard switch on most of my Triumphs, curious yours does not, but it's a Euro bike so may not be required.
