So... is Birth finished or not?
NO, it is not finished.
Why?
MangaUpdates contains information about the anime adaptation and indicates that it is incomplete.
Watching Order (by release date)
Here you have all the anime relations we detected ordered by release date, so you can see the reason for this status in context and also use it as watching order.
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Synopsis
On the once thriving planet Aqualoid, a devastating attack by the Inorganics has left it a post-apocalyptic wasteland.
Our story follows Nam, who unexpectedly stumbles upon a peculiar sword, instantly making him a target of everyone on the planet.
With the relentless Inorganics closing in, Nam and his companions embark on a perilous journey to uncover the secrets of the sword.
Their quest is driven by the hope of saving Aqualoid, while grappling with the unsettling possibility that the sword's true purpose might be something far more destructive, perhaps even a new beginning at the cost of their world.