We extend our previous work by proving that for translation invariant Gibbs states on ${\mathbb Z}^d$ with a translation invariant interaction potential $\Psi=(\Psi_A)$ satisfying $\sum_{A \ni 0}|A|^{-1}[\diam(A)]^d\|\Psi_A\|<\infty$ the following hold: (1) the Kolmogorov-property implies a trivial full tail and (2) the Bernoulli-property implies Følner independence. The existence of bilaterally deterministic Bernoulli Shifts tells us that neither (1) nor (2) is, in general, true for random fields without some further assumption (even when $d=1$).