Bell's theorem proves that local hidden variables theories cannot reproduce quantum mechanics (with the exception of superdeterministic theories). Are there any local theories that do not involve hidden variables? How do such theories explain the observed correlations?
In the context of discussing quantum mechanics, realism means that the evolution of a system can be represented by a number or set of numbers, each of which represents the value of some measurable quantity. Note that this has nothing at all to do with the philosophical idea that reality is objective. It could easily be the case that reality is objective and that the evolution of physical systems can't be described by a set of numbers each of which is measurable.
Quantum mechanics with out modifications like the collapse postulate, can describe the evolution of systems in terms of sets of Hermitian operators, each of whose eigenvalues represents a possible outcome of a measurement. The evolution of those observables is local under the laws of motion that occur in the real world:
The correlations are explained by quantum information being transported in decoherent systems such that the observables of those systems depend on the information being transmitted, but the expectation values of measurements on those observables doesn't depend on that information - locally inaccessible information.
If you include a collapse postulate, or modify quantum mechanics to include collapse, then there is no explanation of how information gets transmitted in measurements of entangled systems.