  We study the stochastic cubic nonlinear wave equation (SNLW) with an additive
noise on the three-dimensional torus $\mathbb{T}^3$. In particular, we prove
local well-posedness of the (renormalized) SNLW when the noise is almost a
space-time white noise. In recent years, the paracontrolled calculus has played
a crucial role in the well-posedness study of singular SNLW on $\mathbb{T}^3$
by Gubinelli, Koch, and the first author (2018), Okamoto, Tolomeo, and the
first author (2020), and Bringmann (2020). Our approach, however, does not rely
on the paracontrolled calculus. We instead proceed with the second order
expansion and study the resulting equation for the residual term, using
multilinear dispersive smoothing.
