A first storage system is configured to participate in a replication process with a second storage system using an active-active configuration. A request for a time-to-live (TTL) grant is received in the first storage system from the second storage system. The first storage system computes an estimate of a difference between local times in the respective first and second storage systems, utilizes the computed estimate in the first storage system to determine a TTL expiration time in the local time in the second storage system, and sends the TTL grant with the TTL expiration time to the second storage system in response to the request. The computed estimate of the difference between the local times in the respective first and second storage systems is illustratively utilized in the first storage system to determine a range for the local time in the second storage system.