You have a bad earth somewhere.
My money is in the connector block un the back of the rear lights. Undo the covers, pull off the connector block - it has a tab you have to depress first (... read it the latest financial figures!
Check that the pins in the light cluster are clean and that the little prongs inside the connector block are "tight". They're like little jaws but can get "weak".
You can clean the prongs by gripping them with a pair of pointy-nose pliers and dragging the pliers off the surface... or use a small file. The "jaws" need squeezing together again gently (don't deform them).
The chances are that only one side is affected. Put the left indicator on and then brake. Listen to see if the indicator stops. Then try the right-hand side.
One side will work and the other won't, I bet. The non-working side is the culprit.
If neither side works, it could be a different problem.. maybe earlier in the circuit.. before the power to the light units diverges.. but less likely. It's one of the connector blocks for sure.