In climate-health impact projection studies, projected impacts from multiple general circulation models (GCMs) are commonly aggregated by reporting the mean of GCM-specific impacts as the point estimate alongside a 95% empirical confidence interval (eCI) constructed from the 2.5th and 97.5th percentiles of the simulated pooled distribution of GCM-specific impacts. This study shows that the eCI generally does not yield the nominal coverage probability for the GCM-ensemble mean and constructs an interval aligned with the estimand. In a simulation study, the coverage of the eCI for the GCM-ensemble mean deviates from the nominal level in both directions, whereas the aligned interval yields coverage near 95% across all considered settings. The exact coverages derived analytically under a location-shift model agree with the simulation results. In a reanalysis of a heat-related mortality projection in London, the eCI is consistently wider. The eCI should be distinguished from confidence intervals for the GCM-ensemble mean; rather, the interval may be better described as a simulation-based approximate prediction interval for a GCM-specific impact under the uniformly randomly selected GCM from the considered GCM set.