  Non-Hermitian Hamiltonians are generally sensitive to boundary conditions,
and their spectra and wave functions under open boundary conditions are not
necessarily predicted by the Bloch band theory for periodic boundary
conditions. To elucidate such a non-Bloch feature, recent works have developed
a non-Bloch band theory that works even under arbitrary boundary conditions.
Here, it is demonstrated that the standard non-Bloch band theory breaks down in
the symplectic class, in which non-Hermitian Hamiltonians exhibit Kramers
degeneracy because of reciprocity. Instead, a modified non-Bloch band theory
for the symplectic class is developed in a general manner, as well as
illustrative examples. This nonstandard non-Bloch band theory underlies the
$\mathbb{Z}_{2}$ non-Hermitian skin effect protected by reciprocity.
