  Barium (Ba), CH, and extrinsic or Tc-poor S-type stars are evolved low- and
intermediate-mass stars that show enhancement of slow-neutron-capture-process
elements on their surface, an indication of mass accretion from a former
asymptotic giant branch (AGB) companion, which is now a white dwarf (WD). Ba
and CH stars can be found in the main-sequence (MS), the sub-giant, and the
giant phase, while extrinsic S-type stars populate the giant branches only. As
these polluted stars evolve, they might be involved in a second phase of
interaction with their now white dwarf companion. In this paper, we consider
systems composed of a main-sequence Ba star and a WD companion when the former
evolves along the Red Giant Branch (RGB). We want to determine if the orbital
properties of the known population of Ba, CH, and S giants can be inferred from
the evolution of their suspected dwarf progenitors. For this purpose, we use
the BINSTAR binary evolution code and model MS+WD binary systems, considering
different binary interaction mechanisms, such as a tidally-enhanced wind
mass-loss and a reduced circularisation efficiency. To explore their impact on
the second RGB ascent, we compare the modelled orbits with the observed period
and eccentricity distributions of Ba and related giants. We show that,
independently of the considered mechanism, there is a strong period cut off
below which core-He burning stars should not be found in binary systems with a
WD companion. This limit is shorter for more massive RGB stars and for more
metal-poor systems. However, we still find a few low-mass short-period giant
systems that are difficult to explain with our models as well as two systems
with very high eccentricities.
