  The crystallographic restriction theorem constrains two-dimensional
nematicity to display either Ising ($Z_{2}$) or three-state-Potts ($Z_{3}$)
critical behaviors, both of which are dominated by amplitude fluctuations.
Here, we use group theory and microscopic modeling to show that this constraint
is circumvented in a $30^{\circ}$-twisted hexagonal bilayer due to its emergent
quasicrystalline symmetries. We find a critical phase dominated by phase
fluctuations of a $Z_{6}$ nematic order parameter and bounded by two
Berezinskii-Kosterlitz-Thouless (BKT) transitions, which displays only
quasi-long-range nematic order. The electronic spectrum in the critical phase
displays a thermal pseudogap-like behavior, whose properties depend on the
anomalous critical exponent. We also show that an out-of-plane magnetic field
induces nematic phase fluctuations that suppress the two BKT transitions via a
mechanism analogous to the Hall viscoelastic response of the lattice, giving
rise to a putative nematic quantum critical point with emergent continuous
symmetry. Finally, we demonstrate that even in the case of an untwisted
bilayer, a critical phase emerges when the nematic order parameter changes sign
between the two layers, establishing an odd-parity nematic state.
