  The tidal tails of stellar clusters provide an important tool for studying
the birth conditions of the clusters and their evolution, coupling, and
interaction with the Galactic potential. We present the N-body evolution of a
Hyades-like stellar cluster with backward-integrated initial conditions on a
realistic 3D orbit in the Milky Way computed within the AMUSE framework. For
the first time, we explore the effect of the initial cluster rotation and the
presence of lumps in the Galactic potential on the formation and evolution of
tidal tails. We show that the tidal tails are not naturally clustered in any
coordinate system. Models with initial rotation result in significant
differences in the cluster mass loss and follow different angular momentum time
evolution. The orientation of the tidal tails relative to the motion vector of
the cluster and the current cluster angular momentum constrain the initial
rotation of the cluster. We highlight the use of the convergent point (CP)
method in searches for co-moving groups and introduce a new compact CP (CCP)
method that accounts for internal kinematics based on an assumed model. Using
the CCP method, we are able to recover candidate members of the Hyades tidal
tails in the Gaia DR2 and eDR3 reaching a total extent of almost 1kpc. We
confirm the previously noted asymmetry in the detected tidal tails. In the eDR3
data we recovered spatial overdensities in the leading and trailing tails that
are kinematically consistent with being epicyclic overdensities and thus
present candidates for the first such detection in an open star cluster. We
show that the epicyclic overdensities are able to provide constraints not only
on the cluster properties, but also on the Galactic potential. Finally, based
on N-body simulations, a close encounter with a massive Galactic lump can
explain the observed asymmetry in the tidal tails of the Hyades.(abriged)
