A multi-chip module (MCM) is described. This MCM includes multiple sites, where a given site in the multiple sites includes multiple chips with proximity connectors that communicate information through proximity communication within the MCM via multiple components associated with the given site. Note that the MCM includes global redundancy and local redundancy at the given site. In particular, the global redundancy involves providing one or more redundant sites in the multiple sites. Furthermore, the local redundancy involves providing one or more redundant chips in the multiple chips and one or more redundant components in the multiple components.