  We construct the exact solution of a non-Hermitian $\mathscr{PT}$-symmetric
isotropic Heisenberg spin chain with integrable boundary fields. We find that
the system exhibits two types of phases we refer to as $A$ and $B$ phases. In
the $B$ type phase, the $\mathscr{PT}$- symmetry remains unbroken and it
consists of eigenstates with only real energies, whereas the $A$ type phase
contains a $\mathscr{PT}$-symmetry broken sector comprised of eigenstates with
only complex energies and a sector of unbroken $\mathscr{PT}$-symmetry with
eigenstates of real energies. The $\mathscr{PT}$-symmetry broken sector
consists of pairs of eigenstates whose energies are complex conjugates of each
other. The existence of two sectors in the $A$ type phase is associated with
the exponentially localized bound states at the edges with complex energies
which are described by boundary strings. We find that both $A$ and $B$ type
phases can be further divided into sub-phases which exhibit different ground
states. We also compute the bound state wavefunction in one magnon sector and
find that as the imaginary value of the boundary parameter is increased, the
exponentially localized wavefunction broadens thereby protruding more into the
bulk, which indicates that exponentially localized bound states may not be
stabilized for large imaginary values of the boundary parameter.
