Cooperation in collective action dilemmas usually breaks down in the absence of additional incentive mechanisms. This tragedy can be escaped if individuals have the possibility to invest in prosocial reward funds that are shared exclusively among cooperators. Yet, the presence of defectors who do not contribute to the public good, but do reward themselves deters cooperation in the absence of additional countermeasures. Although a recent simulation study suggests that interactions in spatially structured populations are sufficient to prevent such antisocial rewarding from deterring cooperation, the exact role of spatial structure on the evolution of public goods cooperation with pool-rewarding remains unclear. Here, we develop an analytical model of public goods with prosocial and antisocial pool rewarding in spatially structured populations. We show that increasing reward funds facilitates the maintenance of prosocial rewarding but prevents its evolution from rare in spatially structured populations, and that spatial structure can sometimes hinder, rather than promote, the evolution of prosocial rewarding. This is due to the fact that, counterintuitively, antisocial rewarding can be a more cooperative alternative than prosocial rewarding in some instances. Our results suggest that, even in spatially structured populations, additional mechanisms are required to prevent antisocial rewarding from deterring cooperation in public goods dilemmas.