In "The Natural Ontological Attitude," Arthur Fine proposes a view distinct from both realism and anti-realism.
Critique of Arguments for Realism. The first part of the paper focuses on arguments for realism, arguing that they are not successful.
I. Ground-Level Arguments. The basic form of these arguments is: Science is incredibly successful. The best explanation of this success is that scientific theories are a literally accurate description of entities that really exist (i.e. scientific realism, in short). So we have good reason to accept scientific realism.
Fine does not actually criticize these ground-level arguments in his essay. Instead, he refers the reader to an essay by Larry Laudan (see Laudan, "A Confutation of Convergent Realism," in Leplin, ed., Scientific Realism; also in Curd & Cover, 1114-1135). This is a complex essay, but the overall picture is fairly straightforward. Laudan argues that this argument simply begs the question against the antirealist. Antirealists like van Fraassen deny that inference to the best explanation gives us a reason to believe in the (realist) truth of the theories it supports. Instead, it only gives us reason to believe in the empirical adequacy of those theories. So it is simply question-begging to use against the antirealist a principle he explicitly rejects!
Laudan also has a second line of criticism of the ground-level argument. Suppose we accept inference to the best explanation as a guide to truth (for the sake of the argument). Realism isn't the best explanation of scientific success unless antirealism lacks an equally plausible explanation. But in fact (the response goes), antirealism has a perfectly plausible explanation of scientific success: it is what scientific theories have been selected for. Scientific theories are evaluated by their success in making predictions, enabling us to control nature, etc. We dump less successful theories in favor of more successful ones. So it's no wonder that the ones we are left with are highly successful!
2. Methodological arguments. These are like the ground-level arguments, except that they appeal to scientific methodology rather than to scientific results. The methods of science are highly successful at leading us to successful theories. The best explanation of the success of scientific methodology is that it leads to theories that are true (in the realist sense). So we have reason to believe in scientific realism.
A generic point is that these second-order arguments apparently depend on the soundness of the ground-level ones. But Fine also makes more specific objections to specific arguments for realism.
A. The problem of the small handful. [Aside: I'm not sure whether this is deliberate or not, but one is tempted to call this the "problem of the few," which would provide a nice contrast with Plato's famous "problem of the many."] In general, science progresses by considering a small number of theories that are very similar to the one being replaced. Why does this work?
Realism's answer: it works because the original theory was approximately true. Since its successors are very similar to the original, they are approximately true also. So, since truth leads to success, we should expect the very similar successors to also be successful.
Fine's criticism: We need to distinguish between (a) success in the same predictions made by the original theory, and (b) success in novel predictions. The antirealist has a perfectly good explanation of (a): we kept the part of the theory that was previously successful, so of course the new one is also successful over the same territory. But it's not clear that we need an explanation of (b). Most tinkering with our theories leads us to theories that are less successful, not more successful. The only ones we keep, though, are the successful ones.
B. The problem of conjunctions. Sometimes we can take two well-confirmed theories and use them in conjunction with one another to generate new predictions that we could not get from either of the theories by itself. Why should we expect these predictions to turn out to be correct? The realist says: because if both theories are true, then their conjunction is also true, and of course true theories will generate true predictions. However, it seems that this explanation is not available to the antirealist.
Fine replies that while this might work for plain old truth, it doesn't work for approximate truth: one would expect the conjunction of two approximately true theories to be less approximately true. So if there's a puzzle about why conjoining theories works, it's a puzzle for realism as well as for antirealism.
Critique of Arguments for Anti-Realism.
There really aren't any criticisms of arguments for antirealism in this article, which may make it look a bit one-sided for a piece which is supposed to be steering a middle way between realism and antirealism. However, Fine has a separate, companion piece, "And Not Anti-Realism Either," in which he criticizes arguments for antirealism.
Defense of an alternative view: NOA. Fine suggests that realists and anti-realists by and large agree in taking both unobservable and everyday objects equally seriously. Somewhat controversially, he uses the term "accept as true" for this kind of "taking seriously." With this understanding of "true," which neither realists nor anti-realists are likely to accept, he can describe the core view they share as an acceptance of science as true. Where realists and anti-realists disagree, on his view, is over the interpretation of truth. Realists add to the core attitude that science is true "a desk-thumping, foot-stomping shout of "Really!"" -- or, a bit more cognitively, an analysis of truth as correspondence to an independently existing reality. Anti-realists, on the other hand, want to offer a different kind of analysis of "true" -- as coherence, or as the set of views scientists would converge on in the ideal limit of inquiry, or something of the sort. (Fine is able to fit van Fraassen into this scheme only by describing van Fraassen's notion of empirical adequacy as a proposed redefinition of "true", which of course is not the way van Fraassen himself would describe his view.) Fine himself suggests that there is no need to add either a realist or an anti-realist theory of truth to the core position -- the core position, i.e. the Natural Ontological Attitude or NOA, is an attractive position in its own right, and the rest is just unnecessary philosophical overlay.