  We consider the Cauchy problem for the defocusing cubic nonlinear
Schr\"odinger equation (NLS) on the waveguide manifold
$\mathbb{R}^3\times\mathbb{T}$ and establish almost sure scattering for random
initial data, where no symmetry conditions are imposed and the result is
available for arbitrarily rough data $f\in H^s$ with $s\in\mathbb{R}$. The main
new ingredient is a layer-by-layer refinement of the newly established
randomization introduced by Shen-Soffer-Wu \cite{ShenSofferWu21}, which enables
us to also obtain strongly smoothing effect from the randomization for the
forcing term along the periodic direction. It is worth noting that such
smoothing effect generally can not hold for purely compact manifolds, which is
on the contrary available for the present model thanks to the mixed type nature
of the underlying domain. As a byproduct, by assuming that the initial data are
periodically trivial, we also obtain the almost sure scattering for the
defocusing cubic NLS on $\mathbb{R}^3$ which parallels the ones by Camps
\cite{Camps} and Shen-Soffer-Wu \cite{Shen2022}. To our knowledge, the paper
also gives the first almost sure well-posedness result for NLS on product
spaces.
