I am not sure whether it would be possible to fix, since backface culling is calculated from the player's location; so F5 mode makes sense that it would not work correctly.

Water is rendering incorrectly in F5 mode, with extra water block faces being rendered when they shouldn't.

The second screenshot is showing the faces of the water behind the flowing water because the player character itself is past that block boundary, so in first person those faces should be exposed as they are.