You already know the root of the problem. Work your way from there...
1- Get the products owners (PO)/business...and let them know the situation.
2- With the POs, divide the departments into two backlog lists. Group them in a way so that the dept. that are most related to each other are bunched up. So now you have 2 backlogs.
3- POs will then re-prioritize based on what is best for the business as one entity and not on what is best for the department.
4- Divide the team. 2 devs and 2.5 devs, each will take care of one backlog.
5- If the two teams need to still know what the others do, apply scrum or scrums.
Just by doing that, you will improve the quality because devs will have a smaller scope to focus on.
