As of iOS 11.3 there is serviceworker feature supported
Even though it has no support for notification. I wonder if it possible to just run any realtime listener, su
Unfortunately not, while Safari on native IOS is getting closer to the functionality we see in chrome, firefox etc. we currently ( as of this comment ) do not have anyway of sending push notifications on native IOS Safari ( or any other ios browser like chrome etc. ).
Safari for mac uses a custom mac solution for these notifications which you can check out here https://developer.apple.com/notifications/safari-push-notifications/.