Omni Supervisor or Chat Supervisor tab missing from Console app

1. Tab Visibility - If a tab is "hidden" from your User Profile, you will not see the tab, even if it is included in a custom app/console that you are viewing. The tab visibility settings need to either be "Default Off" or "Default On". This must be done to either the assigned Profile or an assigned Permission Set.
See also: Tab Settings


2.  Customized Tabs - A User has modified their tab settings (Classic) or their navigation bar (Lightning), and removed the Omni or Chat Supervisor page.

In Classic, this can be corrected by going to the App under Setup and editing it to check the "Overwrite users' personal custom app customization" checkbox [this reverts to the desired tab display to what it is set to on the App]. 
In Lightning Experience, users can customize the navigation bar by adding or removing items, but users can't remove the items Admins added. So if the System Admin added the Omni Supervisor or Chat Supervisor tab to the Console users shouldn't be able to remove it. Admins can also prevent users from customizing their navigation bar for a specific app or for all apps.