  We give a signed fundamental domain for the action on
$\mathbb{R}^{r_1}_+\times{\mathbb{C}^*}^{r_2}$ of the totally positive units
$E_+$ of a number field $k$ of degree $n=r_1+2r_2$ which we assume is not
totally complex. Here $r_1$ and $r_2$ denote the number of real and complex
places of $k$ and $\mathbb{R}_+$ denotes the positive real numbers. The signed
fundamental domain consists of $n$-dimensional $k$-rational cones $C_\alpha$,
each equipped with a sign $\mu_\alpha=\pm1$, with the property that the net
number of intersections of the cones with any $E_+$-orbit is 1.
  The cones $C_\alpha$ and the signs $\mu_\alpha$ are explicitly constructed
from any set of fundamental totally positive units and a set of $3^{r_2}$
"twisters", i.e. elements of $k$ whose arguments at the $r_2$ complex places of
$k$ are sufficiently varied. Introducing twisters gives us the right number of
generators for the cones $C_\alpha $ and allows us to make the $C_\alpha$ turn
in a controlled way around the origin at each complex embedding.
