  We study the phase diagram of two-flavor massless two-color QCD (QC$_2$D)
under the presence of quark chemical potentials and imaginary isospin chemical
potentials. At the special point of the imaginary isospin chemical potential,
called the isospin Roberge--Weiss (RW) point, two-flavor QC$_2$D enjoys the
$\mathbb{Z}_2$ center symmetry that acts on both quark flavors and the Polyakov
loop. We find a $\mathbb{Z}_2$ 't Hooft anomaly of this system, which involves
the $\mathbb{Z}_2$ center symmetry, the baryon-number symmetry, and the isospin
chiral symmetry. Anomaly matching, therefore, constrains the possible phase
diagram at any temperatures and quark chemical potentials at the isospin RW
point, and we compare it with previous results obtained by chiral effective
field theory and lattice simulations. We also point out an interesting
similarity of two-flavor massless QC$_2$D with $(2+1)$d quantum
anti-ferromagnetic systems.
