  We investigate the properties of multidimensional parity-time symmetric
periodic systems whose non-Hermitian periodicity is an integer multiple of the
underlying Hermitian system's periodicity. This creates a natural set of
degeneracies which can undergo thresholdless $\mathcal{PT}$ transitions. We
derive a $\mathbf{k} \cdot \mathbf{p}$ perturbation theory suited to the
continuous eigenvalues of such systems in terms of the modes of the underlying
Hermitian system. In photonic crystals, such thresholdless $\mathcal{PT}$
transitions are shown to yield significant control over the band structure of
the system, and can result in all-angle supercollimation, a
$\mathcal{PT}$-superprism effect, and unidirectional behavior.
