We consider current (1971-2000) and future (2041-2070) average seasonal surface temperature fields from two regional climate models (RCMs) driven by the same atmosphere-ocean general circulation model (GCM) in the North American Regional Climate Change Assessment Program (NARCCAP) Phase II experiment. We analyze the difference between future and current temperature fields for each RCM and include the factor of season, the factor of RCM, and their interaction in a two-way ANOVA model. Noticing that classical ANOVA approaches do not account for spatial dependence, we assume that the main effects and interactions are spatial processes that follow the Spatial Random Effects (SRE) model. This enables us to model the spatial variability through fixed spatial basis functions, and the computations associated with an ANOVA of high-resolution RCM outputs can be carried out without having to resort to approximations. We call the resulting model a spatial two-way ANOVA model. We implement it in a Bayesian framework, and we investigate the variability of climate-change projections over seasons, RCMs, and their interactions. We find that projected temperatures in North America are credibly higher, that the associated warming effects differ in spatial areas and in seasons, and that they are of much larger magnitude than the variability between RCMs.