design – Help me with this sticky double bar in Mobile

I have encountered a UX problem in a site I am designing. I feel there is an obvious solution but I have trouble finding it.

I have a site that has a global sticky bar.

In the middle of the site, I have a tabbed browsing component.

There are 3 tabs and the windows of each one contain a kind of extensive content.

I feel that my users will want to be able to navigate between these tabs without having to slide the page again.

Obviously I do not want to duplicate horizontal sticky bars.

I can not replace or edit the global bar at all.

I'm thinking of a collapse to the side option. Or maybe turn it into a carousel with sticky arrows left / right?

Here is a quick summary of the problem: