Thanks for your replies!
The step response is actually rock solid, no overshoot or oscillation to speak of, even once I start to add capacitive load to the output (that plot was an example of it). That's what confused me, because I was expecting to see ringing and overshoot too.
I think I am beginning to understand why. Kleinstein is right. I have no local feedback around my op amp. All my feedback is coming from the output stage. The result is that the output impedance is very very low (10 mOhm) right up to about 1kHz, so it's very hard to excite any resonance. The impedance only rises to peak at the unity gain point. Makes sense now.
I have learned a lot from this whole process and how open loop response, closed loop response, output impedance etc all interact. It's like poking a bag of water around.
All of the material I read always talk about phase margin and sometimes gain margin, never about what happens inbetween. And as to non-linear modes I guess there is no analytical way to know what will happen, so can only do a lot of testing.