  The formalism recently introduced in arXiv:1610.08468 allows one to assign a
regularity structure, as well as a corresponding "renormalisation group", to
any subcritical system of semilinear stochastic PDEs. Under very mild
additional assumptions, it was then shown in arXiv:1612.08138 that large
classes of driving noises exhibiting the relevant small-scale behaviour can be
lifted to such a regularity structure in a robust way, following a
renormalisation procedure reminiscent of the BPHZ procedure arising in
perturbative QFT. The present work completes this programme by constructing an
action of the renormalisation group onto a suitable class of stochastic PDEs
which is intertwined with its action on the corresponding space of models. This
shows in particular that solutions constructed from the BPHZ lift of a smooth
driving noise coincide with the classical solutions of a modified PDE. This
yields a very general black box type local existence and stability theorem for
a wide class of singular nonlinear SPDEs.
