We don't have them round here, but I have heard roofers say that the synthetic rubber membrane is the best treatment. It might be necessary to lift the lowest row of tiles and put in an apron (don't know what as haven't seen one) to ensure that the water runs off into the middle of the trough and does not wet the wall. The original felt may have rotted.
If you are in an area with these gutters, try to get recommendations. There will be local people who do a lot of it, some will be better than others.
You can't do a proper job off a ladder, it will need scaffold.