You need to complete all quests that are black diamonds with gold exclamation points on the following islands:
Simple as that.
Small black diamond quests are not counted.
Message boards are not counted.
Naval contracts are not counted.
A couple of the quests get split into 'Support Quests', which no longer display as black diamonds with gold exclamation points.
When this happens, they display as small, white diamonds.
They will still appear in your quest list in the pause menu.