This is a typical UI situation:
My suggestion would be that in case that full tab names could not fit the screen (... would start appearing) I would start grouping the inactive tabs into menu’s top-level item captions.
The menu looks like
CRM
|- CRM Screen 1
|- CRM Screen 2
|- CRM Screen 3
Logistics
|- Logistics Screen 1
|- Logistics Screen 2
|- Logistics Screen 3
Settings
|- Settings Screen 1
|- Settings Screen 2
|- Settings Screen 3
So if 3 of the tabs belong to top-menu “CRM” and the first one of those is active, 3 tabs to top-menu “Logistics” and 1 belongs to “Settings” we would display
_____________
| CRM Tab 1 | CRM Tab 2 | CRM Tab 3 | Logistics (3) | Settings (1)
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When clicking on Logistics (3) tab the CRM tabs would collapse and Logistics tabs would expand and first of them would get selected. This way the user would jump into the Logistics group and could select one of the other tabs. The expanded group should always be the first tabs, the collapsed tabs would follow, ordered alphabetically. Top level menu items (that in my opinion should not exist) would never get grouped.
___________________
| Logistics Tab 1 | Logistics Tab 2 | Logistics Tab 3 | CRM (3) | Settings (1)
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Clicking on x of the group would close all the tabs inside.
If any of the tabs in the group would be dirty, the group should also display as dirty.
As this would be a UX breaking change, I would be great to introduce a tooltip, that would appear the first time this would happen and show a description to the user “Inactive tabs are now being grouped…”. After closing the user should not see this tooltip anymore.