A statement similar to the Birkhoff theorem holds: these matrices are the extreme points of the set of matrices for which the $\ell_1$ norm of each row and column is $\leq 1$. I denote by $K$ this set.
It is clear that every such matrix is an extreme point in $K$. Conversely, let $A$ be an extreme point in $K$. The $\ell_1$ norm of each row and column must be $1$. Write the entrywise polar decomposition, i.e. write $A$ as an entrywise product $A=U \circ B$ where $B$ is a bistochastic matrix and $U$ a matrix with entries having modulus $1$. The matrix $B$ is an extremal bistochastic matrix: if $B=(B_1 + B_2)/2$ for bistochastic matrices $B_1 \neq B_2$, then the equation $A = (U \circ B_1+U\circ B_2)/2$ contradicts the extremality of $A$. By the usual Birkhoff theorem, $B$ is a permutation matrix and the result follows.