  Using a new XMM-Newton observation, we have characterized the X-ray
properties of the middle-aged radio-quiet gamma-ray pulsar J0357+3205 (named
Morla) and its tail. The X-ray emission from the pulsar is consistent with a
magnetospheric non-thermal origin plus a thermal emission from a hot spot (or
hot spots). The lack of a thermal component from the whole surface makes Morla
the coldest neutron star in its age range. We found marginal evidence for a
double-peaked modulation of the X-ray emission. The study of the 9'-long tail
confirmed the lack of extended emission near the pulsar itself. The tail shows
a very asymmetric brightness profile and its spectrum lacks any spatial
variation. We found the nebular emission to be inconsistent with a classical
bow-shock, ram-pressure dominated pulsar wind nebula. We propose thermal
bremsstrahlung as an alternative mechanism for Morla's tail emission. In this
scenario, the tail emission comes from the shocked interstellar medium (ISM)
material heated up to X-ray temperatures. This can fully explain the peculiar
features of the tail, assuming a hot, moderately dense interstellar medium
around the pulsar. For a bremsstrahlung-emitting tail, we can estimate the
pulsar distance to be between 300 and 900 pc. A pulsar velocity of about 1900
km/s is required - which would make Morla the pulsar with the largest velocity
- and high inclination angles (>70 degrees) are preferred. We propose Morla's
nebula as the first example of a new "turtle's tail" class of
thermally-emitting nebulae associated to high velocity pulsars.
