newPeople who chat with you on iPhone can get notified about your replies: they add the chat to their Home Screen and enable Notify me.
Push notifications worked everywhere except the place many people hold: the iPhone. Apple allows web notifications only for sites added to the Home Screen, so iPhone readers of your chat were silently out of luck.
A push banner from bespoke on a phone, next to a Home Screen icon with an unread badge
People chatting with you on iPhone can now get your replies pushed to them. The flow is one extra step on their side: add the chat to the Home Screen (Safari's Share menu), open it from there, and tap Notify me. From then on your replies arrive like any app's notifications, badge and all, even with nothing open.
The page detects when it is running as a Home Screen app and adapts, so the Notify button only appears where it can actually work.