An apparatus comprises at least one processing device that includes a processor coupled to a memory. The processing device is configured to establish a secure enclave in cloud infrastructure as part of a trust domain, to load the secure enclave with a program for execution in the secure enclave, and to provide the secure enclave with information sufficient to allow the secure enclave to prove to one or more clients of the trust domain that the secure enclave was established within the trust domain. The provided information customizes the secure enclave for the trust domain in a manner that is detectable by the one or more clients of the trust domain. The establishing, loading and providing are illustratively performed at least in part by an orchestrator component that is part of the trust domain and is implemented using a first physical machine that is separate from a second physical machine used to implement the secure enclave.