Short answer: the question puts into sharp focus the fact that there are various definitions of the term "local" and "realist", which are in need of careful disambiguation.
It's not possible to give a thorough treatment, but let's give a quick review of the most common definitions.
What is a local theory?
There are two main meanings given to the term local:
Local in the sense of EPR and Bell.
This means "no spooky action at a distance", i.e. the probabilities for the results of a process/procedure are not dependent on procedures performed far away (at space-like separations).
Local in the sense of no-signaling.
This means no information can be communicated faster than light from one observer to another.
Quantum theory is non-local in the first sense and local in the second. In particular, it's noteworthy that Bohm's pilot wave theory is deterministic and local (in the 2nd sense).
What is a realist theory?
Here there is even more definitional confusion. Let's list a few noteworthy definitions:
- Realist in the sense of preexisting information about measurement results.
This means that the system "knows" the value of any property, such as position, momentum etc. prior to measurement. This corresponds to Einstein's notion of "elements of reality".
Another way to say it would be: measurements of properties have definite numerical outcomes, which follow deterministically from the prior state of the system.
A shorthand would be to say that the theory includes hidden variables. Copenhagen quantum mechanics is not realist, but the pilot wave theory is realist in this sense.
- Realist in the sense of determinism.
You might be wondering how the previous definition is different from determinism. The big difference is that determinism does not presuppose that measurements of properties have definite numerical outcomes.
For example, the many world theory is deterministic, but not realist in the first sense, because measurements do not result in definite numerical outcomes, but instead in the branching of the world, with all numerical outcomes existing simultaneously.
A further distinction is that strictly speaking the first definition does not require that the time evolution between measurements be deterministic, only that measurements be deterministic.
- Realist in the sense of "indistinguishable from realist".
Unlike the first definition, where the theory's formalism must already include hidden variables, according to this definition a theory is realist if its predictions can be reproduced by another theory with hidden variables. According to this definition, Copenhagen quantum mechanics would be realist because its predictions are indistinguishable from those of the pilot wave theory.
Any physical theory can be reproduced by a realist2 (deterministic) theory because any source of randomness in the theory's formalism can be reproduced by a deterministic "simulation" of randomness. Therefore any physical theory that has a notion of definite numerical measurement outcomes (so not the many worlds theory) is realist in this third sense.
- Philosophical realism.
To satisfy this definition, the theory needs to be interpreted as describing the actual reality, as opposed to, for example, our subjective experience or epistemological state.
This is not so much an attribute of the theory itself, but a philosophical position on what a theory is, or an interpretation of the relationship between a given theory and reality.
What is local realism?
With so many definitions of the two separate terms floating around (and I certainly haven't covered them all!), it might seem like the combined term is even more ambiguous. However, the situation is not as dire. There seems to be far less disagreement on the meaning of the combined term.
A local realist theory is typically understood to be a theory which is Bell-local (i.e. local in the first sense) and realist in the first sense (measurements are not "chancy"). A shorthand would be to say that the theory includes local hidden variables.
(One alternative definition in the literature involves, instead of “normal", deterministic local hidden variables, having local but stochastic ones. I won't spend too much time on the explanation, but it's not too hard to show that this definition reduces to the theory just being Bell-local.)
What did Bell prove?
There are two opinions floating around. Bell's own opinion was that he proved that the results of quantum mechanics cannot be reproduced by a local (in the sense of Bell/Einstein, i.e. the first sense) theory.
Others think that he proved that they cannot be reproduced by a local realist theory.
Philosopher Tim Maudlin, among others, agrees with Bell. I also agree (see my other answer and comments below it for more details).
What about the original question, about a toy example of a local theory which is not realist?
Now that we have disambiguated the definitions, it should not be too hard to find such an example once we decide which exact definition we will be using.
If we decide to use the first definitions of both terms, then the Mystery Particle would be a good example of a simple local non-realist theory. This particle would be a local object that doesn't have a definite color before we shine our flashlight on it. Once we do, it randomly "chooses" the color to become/show.
Great reading resources have been provided by @MaximalIdeal and @gIS in the comments below the original question.