  We embed linear and nonlinear parametrisations of beyond standard
cosmological physics in the halo model reaction framework, providing a
model-independent prescription for the nonlinear matter power spectrum. As an
application, we focus on Horndeski theories, using the Effective Field Theory
of Dark Energy (EFTofDE) to parameterise linear and quasi-nonlinear
perturbations. In the nonlinear regime we investigate both a nonlinear
parameterised-post Friedmannian (nPPF) approach as well as a physically
motivated and approximate phenomenological model based on the error function
(Erf). We compare the parameterised approaches' predictions of the nonlinear
matter power spectrum to the exact solutions, as well as state-of-the-art
emulators, in an evolving dark energy scenario and two well studied modified
gravity models, finding sub-percent agreement in the reaction using the Erf
model at $z\leq1$ and $k\leq 5~h/{\rm Mpc}$. This suggests only an additional 3
free constants, above the background and linear theory parameters, are
sufficient to model nonlinear, non-standard cosmology in the matter power
spectrum at scales down to $k \leq 3h~/{\rm Mpc}$ within $2\%$ accuracy. We
implement the parametrisations into ver.2.0 of the ReACT code.
