Is it possible to get into a spin situation while coordinated?
Yes, what you describe is perfectly possible. A stall in banked flight can result in a spin, given you fly the "right" plane.

My first flight in an ASW-20C was late in the afternoon, when most thermals had died down. I got a winch launch to maybe 350 m (the ASW-20 is rather poor in winch launches) and tried to find an updraft. In maybe 270 m I found one, but it was weak and narrow. So I slowed down to tighten the circle. Having flown a Discus all day before, I was not really alerted to watch out for a departure on stall. At some point the lateral stick force faded markedly, and out of the Discus habit I threw the stick into the circle so the inner aileron would deflect downwards to support the stalling wing. Bad choice! This aggravated the incipient stall, the aircraft rolled into the circle and pitched down. While the view from the cockpit before was blue above and green below, it got now all green. I was entering a spin.

The weakening stick force was the result of flow separation over the aileron, and while the Discus airfoil would still provide more lift with more deflection, the ASW-20 airfoil simply went into full stall.

I immediately pushed down and levelled the plane, and went on to land straight away, having lost maybe 100 m. It all took only seconds, and only long after the flight I could piece all details together. My next flight in the ASW-20 was when the thermals were much stronger. I went up to 1000 m and did stall training until I knew how it felt when stall starts and what my options are. I should had done this on the first flight, and do so on all subsequent new types I fly.