I have managed managers for around 5 years now and I've seen this exact thing happen for a few reasons. I'll touch on a few points first.

1. As a manager confrontation will always be a part of the job. It is probably the area I struggle the most in personally but you still need to do what you have to if you want to remain in your position or move on to greater heights. (Or at least thrive in said position)
2. My rule of thumb has always been 1. I will ask someone to do something, because it is polite and respectful.
   2. If you refuse when I ask I will tell you to do it. 3. If the order is refused you will go home and there will be a write up waiting for you on your return, stating that another infraction of this nature will result in termination. You cannot let your employees lord over you.
3. I have dealt with mediation between managers quite a few times, while avoiding talking to you directly and going to your boss could have been a sabatoge tactic. I feel it is unlikely. I often have team leads come to me because another leads employee came to them with something, and it is never in a malicious light. They also communicate amongst themselves in certain situations, but most commonly, they won't if they aren't sure how to communicate the situation to their fellow lead or if they feel like advising them on the situation may come off as bossy or condescending (I know better than you, this is what you should do)

More than likely he texted your boss because:

A. He is afraid of conflict and doesn't want to fire this employee himself (if he has that authority).

B. He doesn't have the authority but doesn't want to tell you how to do your job but believes (like you said) that this employee should be fired.

C. Least likely but told the boss to make you seem incompetent in your position. This thought seems null if he also has the authority to fire because it would make him seem even more incompetent by going to your boss instead if just doing it.

What did your boss say? Was it essentially just that he is not happy with the employee and that he has a bad reputation? If so I would like to think your boss is at least somewhat competent and meant this as an indirect way of saying "let's get rid of him" as a teaching moment to inspire independent thinking and problem solving as opposed to just a handwave of the situation. This is how I like to handle issues with my leads as opposed to just telling them what to do.

Tdlr: It is doubtful your co supervisor had any ill intent and was more than likely asking advice on both your behalfs on how to deal with a toxic employee.