  In this paper, we investigate a cosmological model based on viscous $f(R,T)$
gravity as a potential alternative to dark energy. This model incorporates bulk
viscosity and is analyzed using an effective equation of state. We consider the
simplest specific model, $f(R,T)=R+\lambda T$, where $\lambda$ is a constant.
The exact solution of our viscous $f(R,T)$ cosmological model is derived, and
then we use the combined datasets consisting of 31 $H(z)$ data points and 1701
Pantheon+ SNe data points to determine the best-fit values of the model
parameters. We find good agreement with observations, particularly at higher
redshifts. Our model's behavior, including energy density, pressure with
viscosity, effective equation of state, and deceleration parameter, is
analyzed. It indicates a shift from decelerated to accelerated phases of the
universe's expansion, suggesting that bulk viscosity in the cosmic fluid could
effectively generate the negative pressure necessary for cosmic expansion.
Finally, we explore statefinder diagnostics to differentiate between various
dark energy models, revealing that our model resides in the quintessence
region.
