  A dilute suspension of active Brownian particles in a dense compressible
viscoelastic fluid, forms a natural setting to study the emergence of
nonreciprocity during a dynamical phase transition. At these densities, the
transport of active particles is strongly influenced by the passive medium and
shows a dynamical jamming transition as a function of activity and medium
density. In the process, the compressible medium is actively churned up -for
low activity, the active particle gets self-trapped in a spherical cavity of
its own making, while for large activity, the active particle ploughs through
the medium, either accompanied by a moving anisotropic wake, or leaving a
porous trail. A hydrodynamic approach makes it evident that the active particle
generates a long range density wake which breaks fore-aft symmetry, consistent
with the simulations. Accounting for the back reaction of the compressible
medium leads to (i) dynamical jamming of the active particle, and (ii) a
dynamical non-reciprocal attraction between two active particles moving along
the same direction, with the trailing particle catching up with the leading one
in finite time. We emphasize that these nonreciprocal effects appear only when
the active particles are moving and so manifest in the vicinity of the
jamming-unjamming transition.
