  We study the stochastic viscous nonlinear wave equations (SvNLW) on $\mathbb
T^2$, forced by a fractional derivative of the space-time white noise $\xi$. In
particular, we consider SvNLW with the singular additive forcing
$D^\frac{1}{2}\xi$ such that solutions are expected to be merely distributions.
By introducing an appropriate renormalization, we prove local well-posedness of
SvNLW. By establishing an energy bound via a Yudovich-type argument, we also
prove global well-posedness of the defocusing cubic SvNLW. Lastly, in the
defocusing case, we prove almost sure global well-posedness of SvNLW with
respect to certain Gaussian random initial data.
