Short-term probabilistic solar forecasts are an important tool in decision-making processes in which uncertainty plays a non-negligible role. Purely statistical models that produce temporal or spatiotemporal probabilistic solar forecasts are generally trained individually, and the combined forecasts therefore lack the temporal or spatiotemporal correlation present in the data. To recover the spatiotemporal dependence structure, a copula can be employed, which constructs a multivariate distribution from which spatially and temporally correlated uniform random numbers can be sampled, which in turn can be used to generate the so-called space-time trajectories via the inverse probability integral transform. In this study, we employ the recently introduced ultra-fast preselection algorithm to leverage the spatiotemporal information present in a pyranometer network and compare its accuracy to that of quantile regression forecasts that only consider temporal information. We show that the preselection algorithm improves both the calibration and sharpness of the predictive distributions. Furthermore, we employ four copulas, i.e., (1) Gaussian, (2) Student-t, (3) Clayton, and (4) empirical, to generate space-time trajectories. The results highlight the necessity to rigorously assess the calibration of the space-time trajectories and the correct modeling of the spatiotemporal dependence structure, which we show through techniques introduced in atmospheric sciences. The code used to generate the results in this study can be found at https://github.com/DWvanderMeer/SpaceTimeTrajectories.