Why doesn't LTSpice predict this op-amp oscillation?
There are different models for the LM358 unit. PSpice simulations based on "LM358" result in a phase margin of app. 50...60 deg. But apparently, this is a very simple model.

However, when using the LM358/NS model the margin is slightly negative!
This explains the observed instability during measurements. Hence, external stabilization of the feedback scheme is necessary. 

Compensation: A compensation scheme (series connection R=500...1000 Ohms and C=50...100nF) at the opamp output node provides a phase margin of app. 50 deg. (simulation).