  The interlayer magnetic interaction in bilayer CrI$_3$ plays a crucial role
for its device applications. In this work, we studied the interlayer magnetic
interaction in $\pi/3$-twisted bilayer CrI$_3$ using first-principles
calculations. Our calculations show that the interlayer coupling can be
ferromagnetic or antiferromagnetic depending crucially on lateral shift. The
strongest antiferromagnetic interlayer interaction appears in the
$\bar{A}A$-stacking. The magnetic force theory calculations demonstrate that
such an antiferromagnetic interaction is dominanted by the $e_g$-$e_g$ channel.
Particularly, the interlayer antiferromagnetic interaction is very sensitive to
external pressure. This highly tunable interlayer interaction makes
$\pi/3$-twisted bilayer CrI$_3$ a potential building block for magnetic field
effect transistors and pressure sensors.
