  It has recently been shown that the phenomenologically successful pattern of
cobimaximal neutrino mixing ($\theta_{13} \neq 0$, $\theta_{23} = \pi/4$, and
$\delta_{CP} = \pm \pi/2$) may be achieved in the context of the non-Abelian
discrete symmetry $A_4$, if the neutrino mass matrix is diagonalized by an
orthogonal matrix ${\cal O}$. We study how this pattern would deviate if ${\cal
O}$ is replaced by an unitary matrix.
