  Pulsars out of their parent SNR directly interact with the ISM producing so
called Bow-Shock Pulsar Wind Nebulae, the relativistic equivalents of the
heliosphere/heliotail system. These have been directly observed from Radio to
X-ray, and are found also associated to TeV halos, with a large variety of
morphologies. They offer a unique environment where the pulsar wind can be
studied by modelling its interaction with the surrounding ambient medium, in a
fashion that is different/complementary from the canonical Plerions. These
systems have also been suggested as the possible origin of the positron excess
detected by AMS and PAMELA, in contrast to dark matter. I will present results
from 3D Relativistic MHD simulations of such nebulae. On top of these
simulations we computed the expected emission signatures, the properties of
high energy particle escape, the role of current sheets in channeling cosmic
rays, the level of turbulence and magnetic amplification, and how they depend
on the wind structure and magnetisation.
