So when there is a new message that has appeared, the tile in the browser could indicate some form of icon that a new message has appeared.
If you look at F-chat, when there is a new message in a PM, the browser tab tile indicates the number of unread message. It doesn’t need to be as sophisticated as that, perhaps the word new appearing for now.
The existing notification uses the operating systems built in System Notifications. So, what you are asking is sort of already implemented, but it will show as a OS notification and not as a browser tab notification.
You can activate notifications under the settings “Cog wheel” at the bottom of the “Realm” panel.
Is your suggestion meant as a complement to existing notifications?
I mean something more like 
It happens for the Forum. But not for the chat.
This feature is now implemented as well 
With the addition that, if at least one of the unseen messages are directed (whispers/summons/messages/etc.), the number will show inside [square brackes] instead of (parenthesis).
Try it out to see if it works allright!
Best regards,
Accipiter
This is fantastic! Thank you!
Will test it out.