When Couchbase Lite iOS tries to start replications with a Couchbase server configured with SSL, you get an HTTP 400 Bad Request error: Handshake Failed.
Future Googlers: I answer my own question in an attempt to save you of the headache that I endured. If you're using nginx, the problem is with the configuration file. You need to add web socket support to your configuration file. Restart nginx and relax, while Couchbase Lite iOS happily syncs with your server.