  We study a class of models for neutrino mass matrix in Type-II seesaw with
$A_4$ family symmetry. The resulting neutrino mass matrix can be naturally made
to respect a $\mu-\tau$ exchange plus CP conjugate symmetry (GLS) with the CP
violating phase $\delta$ and the mixing angle $\theta_{23}$ predicted to be
$\pm\pi/2$ and $\pi/4$, respectively. When GLS is explicitly broken by complex
Yukawa couplings, the model predictions for $\delta$ and $\theta_{23}$ can be
significantly modified. Should future experiments will indeed determine
$\theta_{23}$ and $\delta_{CP}$ away from the GLS limit values, one then had to
consider models with broken GLS. We study several simple scenarios to show how
the modifications arise when GLS is broken and how future experiments can test
this class of models.
