quantum interpretations - How can you be a realist and not posit an ontological model?

I have a specific technical question about how to formalize models for quantum interpretations.

My question arises from the talk Why I am not a psi-ontologist, by Rob Spekkens at the Perimter Institute, in which he explains in detail the differences between $\psi$-ontic and $\psi$-epistemic models for quantum theory, as described in e.g. this previous question, and, more in depth, in Spekkens and Harrigan's paper Found. Phys. 40, 125 (2010), arXiv:0706.2661.

At the start of the talk (first two or three minutes of the video), Spekkens lays out some of the basic terminology for the formalism, starting with

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  an operational theory, which

  
  

  
  

  just gives you a prescription, an algorithm for calculating the probabilities of different outcomes of some measurement given some preparation.

  
  

  
  

  That is, an operational theory just posits the existence of preparation procedures $\sf P$ and measurement procedures $\sf M$, where

  
  
  
  
  • to each preparation procedure $\sf P$ quantum theory associates a density operator $\rho_\sf{P}$, and
  
  
  • to each measurement procedure $\sf M$ quantum theory associates a set $\{x^{(\sf M)}\}$ of possible measurement outcomes, and a set of measurement operators $E_x^{(\sf M)}$ which form a POVM,
  
  
  
  

  and the operational theory restricts itself to talking about the probability $P(x^{(\sf M)}|\sf{P,M})$ of outcome $x^{(\sf M)}$ given a preparation procedure $\sf P$ and a measurement $\sf M$, which is described by quantum theory as $$P(x^{(\sf M)}|\sf{P,M})=\rm{Tr}(\rho_\sf{P} E_x^{(\sf M)}).$$

  
  
  

Spekkens then goes on to talk about realist interpretations:

If you favour realism then you might say that's not enough as an interpretation, and we would also like to be able to say that the system that comes out of this device has some physical properties and it's those physical properties that are relevant to the measurement outcome.

So, I'm going to say that an ontological model of quantum theory is a realist interpretation and it has the following sort of form. It posits ontic states: little lambda is going to denote the physical state of the system that passes between (the preparation and the measurement), and capital lambda is the space of such states.

And, for every preparation procedure you have in the lab, you posit that there is some statistical distribution over physical states of the system that correspond to that preparation procedure.

Then, an audience member (Adrian Kent if I've got it right) intervenes with the following comment:

I would say that to be a realist you don't have to be committed to any ontological model of quantum theory the way you've just defined it. It's a special case of realism.

— to which Spekkens immediately agrees.

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I find this deeply confusing. How can you be a realist as regards quantum theory, and not even posit something that can be interpreted as some big space $\Lambda$ that groups the physical states of the system? What description can be more generic or broad than Spekkens' definition of an ontological model? How do those interpretations look like, and where can I find examples of them used in practice in the literature?