This OoB is possible with all birds, on both islands. You'll want to approach the "turn back" barrier that surrounds the islands. Once you hit it, spam your "flip direction" button, and eventually one of the turn backs will place you on the other side of the trigger, thus letting you freely fly past the borders.

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In some locations you may hit a second turn back barrier, in which case simply repeat the procedure used for the first part

Getting back in is fairly easy. You can simply fly back through the trigger without causing a flip, as you would be now heading in the "right" direction, being back towards the map.

Many portions of the maps, while solid, have holes while flying from the outside in. Likewise, most (but not all) of the ocean floor has only 1-way collisions, meaning it's possible to simply fly up and out back intointo bounds. On the first island, you can also fly up to and touch the portal to the second island, the hitbox of which pokes through the cave walls.