Many of the socio-economic and environmental challenges of the 21st century like the growing energy and food demand, rising sea levels and temperatures put stress on marine ecosystems and coastal populations. This requires a significant strengthening of our monitoring capacities for processes in the water column, at the seafloor and in the subsurface. However, present-day seafloor instruments and the required infrastructure to operate these are expensive and inaccessible. We envision a future Internet of Underwater Things, composed of small and cheap but intelligent underwater nodes. Each node will be equipped with sensing, communication, and computing capabilities. Building on distributed event detection and cross-domain data fusion, such an Internet of Underwater Things will enable new applications. In this paper, we argue that to make this vision a reality, we need new methodologies for resource-efficient and distributed cross-domain data fusion. Resource-efficient, distributed neural networks will serve as data-analytics pipelines to derive highly aggregated patterns of interest from raw data. These will serve as (1) a common base in time and space for fusion of heterogeneous data, and (2) be sufficiently small to be transmitted efficiently in resource-constrained settings.