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)
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