  We give a construction allowing to construct local renormalised solutions to
general quasilinear stochastic PDEs within the theory of regularity structures,
thus greatly generalising the recent results of [BDH16,FG16,OW16]. Loosely
speaking, our construction covers quasilinear variants of all classes of
equations for which the general construction of [Hai14,BHZ16,CH16] applies,
including in particular one-dimensional systems with KPZ-type nonlinearities
driven by space-time white noise. In a less singular and more specific case, we
furthermore show that the counterterms introduced by the renormalisation
procedure are given by local functionals of the solution. The main feature of
our construction is that it allows to exploit a number of existing results
developed for the semilinear case, so that the number of additional arguments
it requires is relatively small.
