Quasi-generalised KPZ equation

We derive the renormalised equation for the quasi-generalised KPZ equation with space-time white noise. We complement the program initiated by Gerencs\'er and Hairer for solving quasi-linear equations using regularity structures by an algebraic machinery that gives a systematic tool to remove non-local counterterms and provide a precise expression of the renormalised equation that is consistent with the semilinear case. The solution theory satisfies the chain rule and a natural notion of It\^o isometry, which can be combined to obtain global in time solution.