We prove the existence of global-in-time regular solutions to a system of stochastic quadratic reaction-diffusion equations. Global-in-time existence is based on a $L^\infty$-estimate obtained by an approach à la De Giorgi, as in [GoudonVasseur10]. The adaptation of this technique to the stochastic case requires in its final step an $L^2\ln(L^2)$-bound, furnished by an estimate by duality on the entropy inequality, as in [DesvillettesFellnerPierreVovelle07]. In our stochastic context, and similarly to [DebusscheRoselloVovelle2021], we need to use a backward SPDE to exploit the duality technique