  We define for every positive Anosov representation of a nonabelian free group
into $\mathrm{SO}(2n,2n-1)$ a family of $\mathbb{R}^{4n-1}$-valued cocycles
which induce proper affine actions on $\mathbb{R}^{4n-1}$. We construct
fundamental domains in $\mathbb{R}^{4n-1}$ bounded by generalized crooked
planes for these affine actions, and deduce that the quotient manifolds are
homeomorphic to handlebodies.
