In climate science regime transitions include abrupt changes in modes of climate variability and shifts in the connectivity of the whole system. While important, their identification remains challenging. This paper proposes a new framework to investigate regime transitions and connectivity patterns in spatiotemporal climate fields. Firstly, local regime shifts are quantified by means of information entropy; secondly, their spatial heterogeneity is examined by identifying the underlying spatial domains of the entropy field; finally, a weighted, direct and time-dependent network is inferred to capture the linkages between these domains. The spatiotemporal variability in sea surface temperature (SST) in two simulations of the last 6000 years is investigated with the proposed approach. The largest regional regime shifts emerge as abrupt transitions from low to high-frequency SST oscillations, or vice versa, in both simulations. Furthermore, the variability in time of the two climate networks is studied in terms of their network density. Generally, rapid and sudden transitions in the degree of connectivity of the system are observed in both simulations but, in most cases, at different times, with few exceptions. This suggests that our ability to predict the climate system may be hampered by its inherent complexity resulting from internal variability.