 Solution

For every $v\in {\mathbb{Z}}^3$, sample independent $U_v\sim \mathrm{Unif}[0,1]$ and set ${\eta }_p(v)=1_{\{U_v\le p\}}$. Then each ${\eta }_p$ has the required Bernoulli product law and ${\eta }_p\le {\eta }_{p^{\prime }}$ whenever $p\le p^{\prime }$. This is the monotone coupling of Bernoulli percolation.