Politics within Project Teams

I worked on a project that became very political. The product owner was a manager and an end user of the system we were building, but didn't work in our organization and had no accountability for budget. They soon began treating the project team members not like colleagues, but as underlings. They insisted that we add several user reps to the project team to help with software testing. Soon, the expanded team began behaving more like a clique than a professional project team. I resolved the situation for me by retiring (the time was ripe). But the project has been trundling on for over a year and a half, over budget and failing to deliver.

It's a typical Matrix structure that has been defined with the Project resources reporting to their respective line managers and the Project Manager having the responsibility for the overall project execution without being the line manager for the project resources.

There is nothing revolutionary that the PM can do in this instance rather than using influence and focusing the team's participation towards the achievement of the common project goal - which is to ensure project success and a product that meets the stakeholder's expectations.
The PM can inspire and keep reminding the team of the Project's business case and how important their individual contributions are towards that big picture and also be appreciative of the efforts of each team member in the project.

The PM can be a servant leader and get "out of the way" of team members , let them do their jobs to the best of their abilities and encourage team members to take the "lead" on particular work packages and to promptly inform them where there are blockers to progress.

I have come across such politics and situations before and the best approach I can suggest is to request team members to stick to schedules and tasks under their realm of responsibilities and to encourage them to escalate if they are unable to achieve those within the stipulated time.
I then either have a conversation with their line manager on how to tackle the problem at hand without blaming anyone and failing that, I approach my project sponsor.

For me is simple to manager this type of situations. The project is to create a solution they need to achieve an organizational objective/goal. So, if they do not think that I help them to understand the reasons and I help them to go to their managers to find explanations or alignement. I put all clear at project kickoff no matter I worked before from the very moment I create the project stakeholder analysis.