This is however non-physical for the positive range, right?
Yeah, this is my understanding.
If there exist a negative energy solution of the BSE (which, by the way, is usually a hint that you are starting from the wrong GS), then you should have a negative peak in the positive energy part of absorption if the anti-resonant contribution is included.
Notice that the peak is negative if the absorption is constructed from the "retarded" four point propagator.
If the T-ordered four points propagator were instead used, than the peak intensity would be positive.
However the physical quantity is the retarded function. The T-ordered is just a mathematical tool needed for deriving equations.
I would say that there is also a straightforward physical interpretation to this. If you have a negative energy solution, you can think that what you are considering as the GS is instead an excited state.
The system can emit light to reach the lower energy ground state. The dielectric function contains both absorption and stimulated emission (we usually call it "absorption" just because the emission channel is always zero from the ground state). Then, if light is emitted, the dielectric function must be negative at that energy.
Best,
D.