  We establish a coupling between the $\mathcal{P}(\phi)_2$ measure and the
Gaussian free field on the two-dimensional unit torus at all spatial scales,
quantified by probabilistic regularity estimates on the difference field. Our
result includes the well-studied $\phi^4_2$ measure. The proof uses an exact
correspondence between the Polchinski renormalisation group approach, which is
used to define the coupling, and the Bou\'e-Dupuis stochastic control
representation for $\mathcal{P}(\phi)_2$. More precisely, we show that the
difference field is obtained from a specific minimiser of the variational
problem. This allows to transfer regularity estimates for the small-scales of
minimisers, obtained using discrete harmonic analysis tools, to the difference
field. As an application of the coupling, we prove that the maximum of the
$\mathcal{P}(\phi)_2$ field on the discretised torus with mesh-size $\epsilon >
0$ converges in distribution to a randomly shifted Gumbel distribution as
$\epsilon \rightarrow 0$.
