We consider energy sub-critical defocusing nonlinear wave equations on R³ and establish the existence of unique global solutions almost surely with respect to a unit-scale randomization of the initial data on Euclidean space. In particular, we provide examples of initial data at super-critical regularities which lead to unique global solutions. The proof is based on probabilistic growth estimates for a new modified energy functional. This work improves upon the authors' previous results (Comm. Partial Differential Equations, 2014) by significantly lowering the regularity threshold and strengthening the notion of uniqueness.

The first author was supported in part by the Swiss National Science Foundation under grant SNF 200020-159925. The second author was supported in part by the U.S. National Science Foundation grant DMS-1362509.