I get your vent. It's painful to deal with people who take over on your call, who talk to your patients, and who don't seem to work as a team. I am part time at my current job, and work with a new person every time.
FWIW, that day is in the books. There's no good way to work on what happened that day besides being proactive in the future.
Next time, first thing in the morning go in with them, hang out chatting till whatever time the boss would say you need to be out checking out the truck, and then say something along the lines of "well, I'm going to head out and check the truck. You coming?" Include in this all tact and politeness you can muster, with a joke or a question for the person, like "I couldn't find the _____, can you give me a hand"
Then, when it's just you two and you're doing all the good morning stuff, work into "do you want to drive or ride first?" If he says he doesn't care, say you want to ride first. Then on the way into the first call, when you're pulling up on scene, take the reins on the communication: "I'll take the clipboard, can you please grab the bag?" When/if he steps on your toes with the interview, just say something at a later time, like "it messes with the flow of things when you jump in with your questions. When it's my call, can you please just use a couple of words to prompt me down the path of whatever you are thinking instead of talking directly to the patient?"
It's a mess when two medics who do not have perfect cohesion are trying to both be the AIC on a call. I have been able to break the "don't talk to the patient on my call" rule with only a couple of partners over the course of my career: my husband and another couple of partners that I really clicked with. We knew where we were headed with the questioning, so we weren't a distraction to the process, we were helping. Even so, even with those few best partners I still primarily keep my mouth SHUT on their calls.
This is not arrogance, it's mutual respect: you do your thing without my interference, then you let me do mine. It only happens when people consciously set the ground rules. It is NOT helpful, only chaotic, to have more than one person talking to the patient. It does not inspire confidence, it looks like there's a subordinate or trainee situation going on, and it's just not standard.