Many environmental processes exhibit weakening spatial dependence as events
become more extreme. Well-known limiting models, such as max-stable or
generalized Pareto processes, cannot capture this, which can lead to a
preference for models that exhibit a property known as asymptotic independence.
However, weakening dependence does not automatically imply asymptotic
independence, and whether the process is truly asymptotically (in)dependent is
usually far from clear. The distinction is key as it can have a large impact
upon extrapolation, i.e., the estimated probabilities of events more extreme
than those observed. In this work, we present a single spatial model that is
able to capture both dependence classes in a parsimonious manner, and with a
smooth transition between the two cases. The model covers a wide range of
possibilities from asymptotic independence through to complete dependence, and
permits weakening dependence of extremes even under asymptotic dependence.
Censored likelihood-based inference for the implied copula is feasible in
moderate dimensions due to closed-form margins. The model is applied to
oceanographic datasets with ambiguous true limiting dependence structure.