  Many binary stellar systems in which the primary star is beyond the
asymptotic giant branch (AGB) evolutionary phase show significant orbital
eccentricities whereas current binary interaction models predict their orbits
to be circularised. We analyse how the orbital parameters in a system are
modified under mass loss and mass exchange among its binary components and
propose a model for enhanced mass-loss from the AGB star due to tidal
interaction with its companion, which allows a smooth transition between the
wind and Roche-lobe overflow mass-loss regimes. We explicitly follow its effect
along the orbit on the change of eccentricity and orbital semi-major axis, as
well as the effect of accretion by the companion. We calculate timescales for
the variation of these orbital parameters and compare them to the tidal
circularisation timescale. We find that in many cases, due to the enhanced mass
loss of the AGB component at orbital phases closer to the periastron, the net
eccentricity growth rate in one orbit is comparable to the rate of tidal
circularisation. We show that with this eccentricity enhancing mechanism it is
possible to reproduce the orbital period and eccentricity of the Sirius system,
which under the standard assumptions of binary interaction is expected to be
circularised. We also show that this mechanism may provide an explanation for
the eccentricities of most barium star systems, which are expected to be
circularised due to tidal dissipation. By proposing a tidally enhanced model of
mass loss from AGB stars we find a mechanism which efficiently works against
the tidal circularisation of the orbit, which explains the significant
eccentricities observed in binary systems containing a white dwarf and a less
evolved companion, such as Sirius and systems with barium stars.
