When the lights are on, all rear bulb are fine. When you press on the breaks, the break lights are fine. When you press on the breaks when the lights are on the right rear goes completely dark.

check the ground wire that is most likely the problem. also on some there is a fuse rare but sometimes there is one .

More than likely its bulbs .

Instructions Open the rear deck lid on the Mazda 6.Carefully remove the plastic finishing cover from the license plate light housing, using a small flat-head screwdriver to release the locking tabs. Pull the cover downward and off of the deck lid, exposing the license plate bulb.
Remove the old light bulb from the license plate bulb socket by hand. Wrap your new one in a paper towel and do not touch the bulb with your bare fingers. Insert the new bulb into the light socket. Install the finishing cover onto the light housing and push it upward until it clicks and locks into place.Repeat Steps 2 and 3 to complete the LED light conversion on the opposite side of the deck lid. Test the new bulbs by turning on the headlights on the Mazda 6.
Mercedes Parts.

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it sounds like a ground problem...sounds like the ground is trying to go through the bulbs to make connection....check all ground wires battery, engine and rear frame...

check the electrical contacts within the bulb fixture for oxidation or damage. Check also the wires and wire harness for damage or connection problems.

change the brake light switch. It could be another problem but this is the most common.

Replace the turn signal switch. Bad contact inside causing a crossing of the circuits. When the turn signal is applied it should open the brake light circuit allowing for just the signal to work. Doesn't sound like the switch is doing it's job.

Check the brake light switch should be a orange wire with 12v then a white wire with 12v when the pedal is pressed.If htat cks good then ck the hazard switch by moving it up and down and wiggeling the knob with the brake pedal pressed if they start working then your multifunction switch is bad and needs to be replaced.

sounds like a bad ground at the socket

Most likely a short in your break line. You might also check the brake switch, but I would think if the break switch was going, it you wouldn't even have the third lite working...