  Using the (2+1)-dimensional viscous hydrodynamic code VISH2+1 with a
temperature dependent specific shear viscosity (eta/s)(T), we present a
detailed study of the influence of a large hadronic shear viscosity and its
corresponding relaxation time tau_pi on the transverse momentum spectra and
elliptic flow of hadrons produced in 200 A GeV Au+Au collisions. Although
theory, in principle, predicts a well-defined relation T*tau_pi =
kappa(T)*(eta/s)(T), the precise form of kappa(T) for the matter created in
relativistic heavy-ion collisions is not known. For the popular choice kappa=3
the hadron spectra are found to be insensitive to a significant rise of eta/s
in the hadronic stage, whereas their differential elliptic flow v_2(p_T) is
strongly suppressed by large hadronic viscosity. The large viscous effects on
v_2 are strongly reduced if (as theoretically expected) kappa(T) is allowed to
grow with decreasing temperature in the hadronic stage. This implies that,
until reliable calculations of kappa(T) become available, an extraction of the
hadronic shear viscosity from a comparison between VISH2+1 and a microscopic
hadron cascade or experimental data requires a simultaneous fit of (eta/s)(T)
and kappa(T).
