Why does elevator input move the turn coordinator ball in steep turns? When practicing steep turns, I'm running into an unusual phenomenon.  If the airplane is losing altitude in coordinated flight and I apply aft stick pressure to pitch up and correct, I notice that the ball swings to the outside of the turn indicating a skidding condition.  Conversely if the airplane is gaining altitude and I release back pressure to the stick to descent, the ball always swings to the inside of the turn, indicating a slip.  This seems counter intuitive as one would think that an increase in AoA to gain altitude would result in increased adverse yaw, requiring more rudder pressure in the direction of the turn and less rudder pressure when the nose is lowered resulting in a lower AoA and adverse yaw.

I suspect the reason here has to do with the direction the nose is being forced in by the elevator input during the turn.  At a steep bank angle the nose is inadvertently being pulled inside the turning flightpath which results in a skidding condition, which should, therefore, be countered by less rudder pressure in the direction of the turn.  Conversely when elevator pressure is reduced, the nose would tend to stray outside of the tangential flightpath, resulting in a slipping condition.  The end byproduct of this would be reduced or even cross control inputs needed to maintain coordinated flight.  Can anyone else confirm this?
In a steep turn you are making power changes, pitch changes and also making constant corrections with aileron to hold the bank angle, without even realizing it.  Once established in the turn some airplanes require a bit of in-turn aileron to hold the bank angle, some hold the bank with neutral aileron, and some require top aileron to keep from overbanking.  The top and down aileron inputs are inducing constantly changing adverse yaw forces which come and go, changing torque effects from the engine with power adjustments are producing changing yaw forces, and gyro precession from the prop from pitch motions are inducing yaw forces which come and go, and there are bumps, and you start running into your own wake.

In other words, you are in a machine pulled along by a big torque producing gyro that is slithering and sliding around in a gas with half a dozen forces and moments interacting on it simultaneously.  With all the subtle forces and inputs happening during a turn, I don't think you can identify and act on a single phenomenon like that.   What you have to do is, well, just do whatever it takes to hold the bank angle, altitude and center the ball and don't overthink it.  

If you fly gliders, which use a yaw string that is more sensitive than a ball, even without the torque and gyro effects of an engine, the yaw string drifts this way and that while in a turn, seemingly independent of the aileron position some of the time. You generally make stabs of rudder in concert with aileron, but sometimes the yaw string seems to have a mind of its own and you just do what you have to do with your feet.