  Context. Open clusters that emerged from the star forming regions as
gravitationally bound structures are subjected to star evaporation, ejection,
and tidal forces throughout the rest of their lives. Consequently they form
tidal tails that can stretch kiloparsecs along the cluster's orbit. Aims.
Cluster members are typically found by searching for overdensities in some
parameter space (positions and velocities or sometimes actions and orbital
parameters of stars). However, this method is not effective in identifying
stars located in the tidal tails far from the open cluster cores. We present a
probabilistic method for finding distant cluster members without relying on
searching for overdensities and apply it to 476 open clusters. Methods. First,
we simulate the dissolution of a cluster and obtain a probability distribution
(likelihood) describing where cluster members are to be found. The distribution
of stars from the Gaia DR3 catalogue in high likelihood regions is then
compared to the simulated stellar population of the Galaxy to define the
membership probability of each star. Results. The survey of cluster members
included all stars with a magnitude of G < 17.5 and larger clusters with an age
of > 100 Myr within 3 kpc from the Sun. We successfully find stars with high
membership probabilities in the tidal tails of most clusters, stretching more
than a kiloparsec from the cluster cores in some cases. We analyse the
morphological properties of tidal tails, demonstrate how properly normalised
membership probabilities aid systematic studies of open clusters and publish a
catalogue of stars found in the tidal tails.
