What is the underdetermination of theories in philosophy of science? Is it the lack of ability to choose between alternative scientific theories? If so, how can it be that more than one theory describes the nature correctly? Would it not contradict realism?
Underdetermination of theory by evidence, explored in great detail by Quine, means that from finitely many observations and measurements, that we are able to make by any point in time, even combined with perfect methodology, we are unable to determine a unique theory consistent with them. In other words, even if there were such a thing as the correct theory of nature we lack physical capabilities to find out what it is for sure, distinct theories may well be empirically equivalent. This is over and above the interpretational indeterminacy, where theories are "mathematically" equivalent despite postulating existence of different kinds of entities. For instance, Copenhagen and Bohmian interpretations of quantum mechanics are interpretationally distinct, but mathematically equivalent. They, or rather their natural extensions, may become inequivalent if in the future certain new types of measurements beyond quantum mechanics will become possible. These presumed extensions are only empirically equivalent for the time being.
As for compatibility with realism, it depends on the type of realism in question. It is generally agreed that indeterminacy rules out the most naive form of realism, which simply projects current theoretical entities onto objective reality. More sophisticated forms of realism impose restrictions on "realistic" theories in addition to their simple agreement with the observations (such as simplicity, theoretical unity), and/or restrict the scope of what is claimed to be objectively real.
Structural realism for instance concedes that "essences" of objects do not enter our experience, and only relations among them are reflected in our theories. In this view interpretational indeterminacy is baked into the cake, as long as there is a transformation connecting two theories that preserves their relational structure, they equally reflect as much of reality as we are entitled to. This can be tightened up by insisting that theories not be considered in isolation but in a holistic system that exhibits internal coherence. A sophisticated realist then admits that even big pieces of successful theories may be discarded (like ether and phlogiston were), but optimistically hopes that as science progresses larger and larger portions of it will match the objective reality.