When a user get an invite to a Realm they see a message icon with red dot on the main menu Realm button. 

Given that the user does not own a Realm they will get a "Try/Buy Realms" overlay/window after they click the Realm button in the main menu. This Try/Buy Realms" overlay/window blocks the rest of the Realm interface and the user can only click Try or Buy from that window or Back button. 

The invite to the Realm is not visible nor clickable.

If the user goes back to the main menu the message icon with the red dot is still on the Realms button.

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I (accidentally) found a workaround for this and made a short video showing it. https://youtu.be/4fx8AGjlzJ0 (unlisted on YT)

The workaround is that the user needs to change the UI to Windowed mode. Click the Realms button and then maximize the Minecraft window. Then the user can finally see the Realm invitation icon in the top part of the window and can click it and accept the invite. 

So a fix to this issue would be to look at the Try/Buy Realms" overlay thing, I guess.