Strengths of Realism
Realism and anti-realism are conventionally presented as dichotomous: you must be either one or the other with nothing in between. This is supposed true across the board, from material objects to moral values. But on reflection the dichotomy is too simple—there are finer distinctions to capture. We can approach the matter by examining the paradigm case: realism and anti-realism about the external world. What is it to be a realist about material objects? Several points might be mentioned: propositions about material objects must be logically independent of propositions about sense experience; material objects must be the cause of sense experiences; material objects cannot be mental constructions of any kind; material objects must differ in their intrinsic nature from sense experiences; material objects must exist in a different space (or region of space) from that occupied by sense experiences; material objects must pre-exist and post-exist sense experiences; material objects must have properties that sense experience does not reveal or perhaps cannot reveal. These points affirm that material objects in no way reduce to or depend upon sense experiences; and they are precisely what is denied by someone who cleaves to an anti-realist view of them. So realism here consists in a conjunction of separate claims that are not necessarily jointly true. Consider Berkley’s idealism: he regards so-called material objects as ideas in the mind of God that can exist whether we have corresponding ideas or not, but he does not suppose that they have an intrinsically different nature from sense experiences, since that is what they are. He also believes they exist in a space separate from that occupied by human minds, but he doesn’t think they pre-exist existence in God’s mind. Nor does he hold that propositions about material objects are logically independent of propositions about God’s mind. So is Berkeley a realist or an anti-realist? The question has no sensible answer: he accepts some of the claims of the cluster I mentioned but not all. It seems right to say that he is not so strong a realist as someone who accepts all the claims of the cluster but that he is also not an outright anti-realist who rejects all of them. We might say (not very illuminatingly) that he is a weak realist about material objects, and then go on to specify exactly what claims he accepts and what he rejects. The traditional dichotomy is just too crude to capture the full range of metaphysical opinion in this case.
Or consider realism and anti-realism about the mind itself. You can hold that there is nothing in the mind except what shows itself in actual behavior; or you can weaken this to maintain that mental states consist in dispositions to behavior; or you can identify mental states with brain states that underlie such dispositions; or you can hold that it must be at least logically possible to manifest a mental state behaviorally. Correspondingly, you can assert that mental states exist in a separate immaterial substance that is logically independent of the body and behavior, or you can weaken this position in various ways. The result is a spectrum of possible positions not a simple dichotomy. Some positions are intuitively more realist than others. The closer the position gets to the analogous position with respect to material objects the more or less realist it becomes (existing in a separate space and having a different intrinsic nature make the position strongly realist). But it is artificial and distorting to try to force a position into one or the other of two categories, realist or anti-realist. Someone might reasonably maintain that he is moderately realist about X but not mad-dog realist about X —soft-core but not heavy-duty. On a realism scale of 1 to 10, he might describe himself as a 7.
Much the same pattern is discernible with respect to mathematical realism. You can be an extreme platonic realist holding that numbers exist in a separate sphere difficult to reach from the human point of view, eternal and unchanging, far from the madding crowd of empirical particulars; or you can weaken this position in various ways, holding (say) that numbers are constructions from sets of particulars combined with logic, or even concrete aggregates of particulars. Again, there is room for manoeuver in articulating a position deserving the name of realism, with some positions stronger than others. An anti-realist might accept nominalism or some form of psychologism, where again different strengths of position might be distinguished (for example, numbers are nothing but actual inscriptions in contrast to possible inscriptions). There is a wide spectrum of possible positions that may be adopted and it would be procrustean to try to force all of them into one of two categories. Similarly with scientific realism: one might hold that unobservable entities are real and causal while also holding that they consist in potentialities not actualities; or one might accept particles as real but jib at fields. There is room for half-hearted scientific realism as well as the full-throated kind.
I have made these points as a preparation for considering moral realism. For here there are difficult questions of formulation and it is helpful to have a clear view of the full range of options. We don’t want to lapse into anti-realism just because we have a limited view of the varieties of moral realism. If we want to keep the analogy with the external world, which gives the issue clarity and bite, we need to identify features of the moral case that match the features I listed earlier—such as intrinsic difference of ontological kind or separation in space. Thus moral values may be said to exist at some remove from the moral subject and to differ in kind from any fact about that subject. They must also pre-exist recognition by the subject and be logically independent of anything she might believe, feel, or experience. Presumably they will not be said to act as causes, but that view is logically available under some ingenious conception of causation. Moral values might exist and yet not be discoverable by moral agents, and they may be quite other than what is generally believed. Again, it is possible to endorse some of these claims but not others: for example, one might hold that what is morally right cannot be inferred from what people believe but that morality must be in principle accessible to moral believers—belief-independent but not completely mind-independent. One might believe in Plato’s Form of the Good or in Moore’s indefinable non-natural property of goodness; but it would also be possible to style oneself a moral realist while rejecting such views, opting instead for a view in which the existence of objective reasons constitutes the sole content of a reasonable moral realism. There is no point in fighting over labels, which is a temptation if the issue is conceived dichotomously; better to accept a plurality of possible views each inviting the label “realism”. Some types of moral realism will be stronger than others, i.e. closer to the paradigm of realism about material objects. To insist that certain views are not really realist is to be in thrall to binary thinking, though no doubt certain views will count as anti-realist if any view does (emotivism, for instance). 
One response to these observations would be to abandon all talk of realism and anti-realism as misleadingly simplistic; and that response is not without its merits. But then there is the question of what might be put in its place—what other terminology could we use? And the current terminology is not without intuitive force, especially in conjunction with the paradigm supplied by the external world. To be a realist is definitely to be an identifiable kind of thing—to adopt an intelligible position. The concept is not empty. It is just that it is not quite as black and white as it has seemed from traditional debates. We need to make room for the partial, qualified, and week-kneed realist—as well as the modest and lukewarm anti-realist. Certainly, we must avoid pinning caricatures on positions that attract the label “realist”, as if anything so called must be of the most extreme and implausible kind.
 In ethics we find a contrast between subjectivism and objectivism, as well as between relativism and absolutism, but we don’t find these contrasts in the case of the external world and other subject matters. It is an interesting question why this is so, but it must surely be connected to the fact that there is a strong tendency in ethics for people to believe that thinking it makes it so, which is not the case for the external world. Thus moral realism is often framed as the denial that moral belief implies moral truth (suitably relativized). I would prefer to label this position “moral objectivism” and keep the label “moral realism” for views that model ethics more closely on the external world: but this is all a matter of words (not that words can’t be philosophically important).