  Open systems with anti parity-time (anti $\mathcal{PT}$-) or $\mathcal{PT}$
symmetry exhibit a rich phenomenology absent in their Hermitian counterparts.
To date all model systems and their diverse realizations across classical and
quantum platforms have been local in time, i.e. Markovian. Here we propose a
non-Markovian system with anti-$\mathcal{PT}$-symmetry where a single
time-delay encodes the memory, and experimentally demonstrate its consequences
with two time-delay coupled semiconductor lasers. A transcendental
characteristic equation with infinitely many eigenvalue pairs sets our model
apart. We show that a sequence of amplifying-to-decaying dominant mode
transitions is induced by the time delay in our minimal model. The signatures
of these transitions quantitatively match results obtained from four, coupled,
nonlinear rate equations for laser dynamics, and are experimentally observed as
constant-width sideband oscillations in the laser intensity profiles. Our work
introduces a new paradigm of non-Hermitian systems with memory, paves the way
for their realization in classical systems, and may apply to time-delayed
feedback-control for quantum systems.
