  The generation and verification of genuine multipartite nonlocality is of
central interest for both fundamental research and quantum technological
applications, such as quantum privacy. To demonstrate genuine multipartite
nonlocality in measurement data, the statistics is commonly postselected by
neglecting undesired data. Until now, valid postselection strategies have been
restricted to local postselection. A general postselection that is decided
after communication between parties can mimic nonlocality, even though the
complete data are local. Here, we establish conditions under which genuine
multipartite nonlocality is demonstrable even if observations are postselected
collectively. Intriguingly, certain postselection strategies that require
communication among several parties still offer a demonstration of genuine
multipartite nonlocality shared between all parties. The results are derived
using the causal structure of the experiment and the no-signalling condition
imposed by relativity. Finally, we apply our results to show that genuine
three-partite nonlocality can be created with independent particle sources.
