[For part I see Nonlinear Anal., Theory Methods Appl. 12, No.9, 895-919 (1988; Zbl 0666.35043). Part II (to appear).] We consider quasilinear reaction-diffusion systems of the form \[ (*)\quad \partial_ tu-\partial_ j(a_{jk}(x,t,u)\partial_ ku)=f(x,u,\partial u) \] together with natural (nonlinear) boundary conditions, where u is vector valued. If the matrices \(a_{jk}\) are of a suitable triangular form and f is affine in \(\partial u\) it is shown that (*) possesses a unique global solution provided an a priori bound in the \(L_{\infty}\)-norm is known to exist. In general, that is, if the \(a_{jk}\) are fully occupied matrices, the same is shown to be true if it is known a priori that some Hölder norm of the solution is bounded.