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.
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