= Solution
Use the <probability space> $([0,1]^{\mathbb Z^d},\mathcal F,\mathbb P)$ with the <product measure> of the <uniform distribution> on $[0,1]$. Its coordinate <random variables> $U_v$ are <independent and identically distributed random variables>. Define simultaneously for every parameter
$$
\boxed{X_p(v)=\mathbf1_{\{U_v\leq p\}},\qquad p\in[0,1],\ v\in\mathbb Z^d}.
$$
For $r\leq s$, the <indicator function> inequality $X_r(v)\leq X_s(v)$ holds pointwise at every <graph vertex>. For fixed $p$, each coordinate has the <Bernoulli distribution> with parameter $p$, and the coordinates are independent because they are functions of distinct independent $U_v$. Thus every marginal configuration is exactly the required <site percolation> model. This is a continuum-indexed family of processes, all on one <probability space>, and is the standard <monotone coupling of Bernoulli percolation>.
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