  Phonon nonreciprocity, indicating different transport properties along
opposite directions, has been observed in experiments under a magnetic field.
We show that nonreciprocal acoustic phonons can also exist without a magnetic
field nor net magnetization. We focus on PT symmetric antiferromagnets that
break both time-reversal T and inversion symmetry P. We identify crucial
contributions in phenomenological elastic theory, dubbed flexo-viscosity and
flexo-torque, that induce phonon nonreciprocity without changing the phonon
polarization. The microscopic origin of these contributions is the molecular
Berry curvature, manifested as emergent nonlocal magnetic fields on phonons.
The symmetry breaking originated from spin order is transferred to the phonon
system through spin-orbit coupling, where the orbital degree of freedom affects
the lattice dynamics directly. By electrically modifying the spin-orbit
coupling, we show that both the phonon nonreciprocity and helicity can be
controlled and enhanced. Importantly, the phonon nonreciprocity is an odd
function of the N\'eel vector, serving as an indicator of the order parameter.
