One general aspect of asynchronous local and remote generation of consistent point-in-time snap copies in consistency groups in accordance with the present description, is directed to receiving a local-remote pair point-in-time snap copy establish command to establish a local point-in-time snap copy in a local data storage system, and a remote point-in-time snap copy in a remote data storage system. In one aspect, establishment of the remote point-in-time snap copy relationship may be delayed as a function of the state of a consistency group formation process upon receipt of the local-remote pair point-in-time snap copy establish command. As a result, consistency of the local-remote pair of point-in-time snap copies may be realized, notwithstanding an asynchronous relationship between the local and remote storage systems. Other features and aspects may be realized, depending upon the particular application.