  We prove that the natural $(\operatorname{Aff}
_2(\mathbf{C}),\mathbf{C}^2)$-structure on an Inoue surface is the unique
$(\operatorname{Bir}(\mathbb{P}^2),\mathbb{P}^2(\mathbf{C}))$-structure,
generalizing a result of Bruno Klingler which asserts that the natural
$(\operatorname{Aff}_2(\mathbf{C}),\mathbf{C}^2)$-structure is the unique
$(\operatorname{PGL} _3(\mathbf{C}),\mathbb{P}^2(\mathbf{C}))$-structure.
