The Casella Berger (2002) solutions manual says that the minimal sufficient statistic for
$$f(x) = e^{-(x-\theta)}e^{-e^{-(x-\theta)}}, \qquad x\in \mathbb{R}$$
are the order statistics $(X_{(1)},\dots,X_{(n)})$. This confuses me because it can be written as
$$f(x) = e^{\theta}e^{-x}e^{-e^\theta{e^{-x}}}$$
Which seems to be in the form of a full-rank exponential family, with complete sufficient statistic $T(X) = \sum_i e^{-X_i}$, and hence $T(X)$ is minimal sufficient by Bahadur's theorem. $T(X)$ seems to achieve a much greater reduction in the data aswell, so that the order statistics cannot be minimal sufficient?
I wonder if there is a property of the PDF I am missing that means it's not actually exponential family?