Unfortunately, exactly how you define realism sometimes depends on whether or not you are trying to disprove it.
For instance you might label a theory as having realism if measurements passively reveal a preexisting property. And there is a lot to unpack there. First we will review some quantum mechanics.
Basically if you measured the $\hat z$ component of a spin 1/2 system twice in a row then the second time you will get the same result as you got the first time.
The actual state after the first measurement is one that must give that result for that kind of measurement (the $\hat z$ component of that spin 1/2 system). You don't have to debate realism for this, it is just what Quantum Mechanics predicts. The state sometimes perfectly and reliably predicts a particular measurement outcome.
So lets now discuss another example. First you measure the $\hat z$ component of a spin 1/2 system and then instead of doing another $\hat z$ measurement you measure the $\hat x$ component of the same spin 1/2 system. Now you get a result of $\pm \hbar/2$ but now you can bring realism in since multiple results are predicted.
One approach is common if you want to disprove realism. You say that realism means the system that just underwent a measurement of the $\hat z$ component of a spin 1/2 system actually has
- a spin of $+\hbar/2$ for a $\hat x$ component of the spin, or
- that it has a spin of $-\hbar/2$ for a $\hat x$ component of the spin.
And furthermore, that a $\hat x$ measurement, if done now, would reveal that property. You need the last part about revealing, since you can't just throw the word actual around without some experimental consequences or else it is meaningless.
And now you can argue that realism makes predictions. And they are predictions that disagree with Quantum Mechanics. But that's exactly how you define realism if your goal is to disagree with Quantum Mechanics.
If you don't want to disagree with Quantum Mechanics, you can still have realism. You just have to say that measurements change the state of the system rather than passively revealing something.
For instance in Bohmian Mechanics they can be realists about position, and then they say that spin measurement outcomes are determined solely by the spin state of the system, the type and calibration of device used, and the position.
So someone using Bohmian Mechanics could say they have realism because they were a realist about enough things to totally determine the results (states and position), but they didn't try to be a realist about other things (like components of spin) besides the things that were enough to determine all the results.
And no one one should try. Because the results you get for different measurements (e.g. two $\hat z$ and an $\hat x$) can depend on the order you do them ($\hat z,$ $\hat z,$ $\hat x$ always have the two $\hat z$ agree with each other, and $\hat z,$ $\hat x,$ $\hat z$ can have the two $\hat z$ disagree with each other). So clearly what we call a measurement is an interaction that changes the state and not a passively revealing of knowledge. It can change a state from an eigenstate of $\hat\sigma_z$ into an eigenstate of $\hat\sigma_x.$ You can not expect noncommuting operators to passively reveal preexisting eigenvalues, that would not make sense for the noncommon eigenvectors.
It's not so different than the colloquial idea that things appear a certain way becasue they already were a particular way and that the correspondence is pretty tight.
In Quantum Mechanics when you have multiple results possible for one state, it's hard to have a tight correspondence. If you add something in addition to a state to make a tight correspondence you can get something just like realism ... if you want. But you can't have more than is needed to determine the results because then you go beyond a tight correspondence to an inconsistent theory.
So Bohmian Mechanics as an example has to stop with states and position and doesn't have spin measurements passively reveal preexisting components of spin. It just has a state and a position.