So in theory I should be able to turn myself with ailerons only in the same
way, without the need to use the rudder to stay coordinated. What's the
secret? Each time I try to turn at the same rate that the AP manages, I have
to use the rudder to stay coordinated.


As Maxwell observes, this may just be a simulator limitation. In
actuality, a small amount of rudder should be applied when initiating
and recovering from a turn, but once in a constant bank turn, the
aircraft will normally be in coordinated flight without rudder
application.