I am a Ph.D. candidate in economics in the UK, and now start my 4th and hopefully final year of the program.

I currently consider changing my supervisor (SV hereafter) because of an ongoing conflict. During my second year, she matched me with a former Ph.D. of hers, "co-author" or "CO" from now on, so the three of us could write a paper together. While things started OK, CO became increasingly possessive of the project. His tone became aggressive and disrespectful, he gave me non-sensical, inconsistent instructions, lied to my supervisor behind my back about things I had never done or said (learned of this during meetings with my supervisor), and he repeatedly deleted content I produced from our draft for reasons that stood in complete contradiction to earlier agreements. Suffice to say, progress was unsatisfying.

I was baffled that SV did not intervene, but seemed to go along with him; I had severe self-doubts. Yet after a particular crappy "revision" of our draft by CO, SV shortly after told me she considers CO's input crap and admitted he's difficult -- turns out she was not paying attention and had until then simply trusted CO by default. I was relieved, but unfortunately, nothing changed afterward; while SV repeatedly agreed with my views, she nevertheless continued to humor CO, and backed him when he criticized me for lack of progress. At some point, SV finally admitted to me she has no understanding of our project topic and what we are doing, which explained a lot of her weird behaviour.

This toxic situation has taken a toll on me mentally, and it needs to stop. Unfortunately, SV is a leader of our rather small field, and I was told not having a letter from her is a red flag in job applications. Fortunately, I have done research assistance for another young prof at our faculty who appreciates me, and he would be willing to take me on as his student.

Any advice on how I can handle this gracefully? I have no reason to trust my SV anymore, and think her perfectly capable of ruining my reputation if I changed. Is there a way to protect me against this? Or should I just accept the academic trail is dead for me and look for suitable industry jobs instead?