  Within the framework of relativistic fluctuating hydrodynamics we compute the
contribution of thermal fluctuations to the effective infrared shear viscosity
of a conformal fluid, focusing on quadratic (in fluctuations), second order (in
velocity gradients) terms in the conservation equations. Our approach is based
on the separation of hydrodynamic fields in soft and ultrasoft sectors, in
which the effective shear viscosity arises due to the action of the soft modes
on the evolution of the ultrasoft ones. We find that for a strongly coupled
fluid with small shear viscosity--to--entropy ratio $\eta/s$ the contribution
of thermal fluctuations to the effective shear viscosity is small but
significant. Using realistic estimates for the strongly coupled quark--gluon
plasma created in heavy ion collisions, we find that for $\eta/s$ close to the
AdS/CFT lower bound $1/(4\pi)$ the correction is positive and at most amounts
to 10% in the temperature range 200--300 MeV, whereas for larger values $\eta/s
\sim 2/(4\pi)$ the correction is negligible. For weakly coupled theories the
correction is very small even for $\eta/s=0.08$ and can be neglected.
