Science assumes the real world in the same sense we assume that the Sun goes around the Earth in our everyday lives, or mathematics assumes an ideal realm populated with numbers and structures. It is a practical attitude of a working scientist (farmer, mathematician,...) that saves time and effort on complications irrelevant to the task at hand. Upon reflection one could conjecture that this attitude does reflect operation in a mind independent world inhabited by real things. A realist might even argue that doing otherwise undermines our usual activities, scientific activities in particular, and leaves them hanging. But this reasoning is moralizing and emotional, not rational. Which brings us directly to what it means to have "faith in science": what is the goal of science?
Plato once taught that the goal of geometry is to lift the soul from the bonds of the sensible to higher pastures of philosophy. In a similar vein a realist might say that it is uncovering the hidden reality of nature that animates science. But this stance naturally undermines itself, once science replaces the apparent reality of everyday life (or older theory) with deeper scientific reality, and transfers its realist commitments to the latter, the same doubt arises about the latter as it raises about the former. Indeed, scientists are trained not to take appearances at face value and seek ever deeper explanations. Cao and Schweber give an interesting account of how this dynamic plays out in modern physics in Conceptual Foundations and the Philosophical Aspects of Renormalization Theory:"the recent developments support a pluralism in theoretical ontology, an antifoundationalism in epistemology and an antireductionism in methodology. These implications are in sharp contrast with the neo-Platonism implicit in the traditional pursuit of quantum field theorists... which assumed that, through rational (mainly mathematical) human activities, one could arrive at an ultimate stable theory of everything." (see especially pp.73-77).
The scientific method itself is not a natural extension of realism, but something in tension with it. The hypothetico-deductive origin of mature scientific ontologies plainly means that they took shape in speculation, only empirical consequences of which were confirmed afterwards. This gives rise to the famous problem of underdetermination of scientific theories, associated with Duhem and Quine. And the "no miracles" argument from empirical success to realism is acknowledged to be logically uncompelling even by realists. Looking at history it is hard to expect that fundamental theories of today can not share the fate of geocentrism and ether, whose empirical consequences are nonetheless fully integrated into the modern theories, affirming the empirical continuity of science.
Anti-realism in ontology goes hand in hand with instrumentalism in epistemology, and a different understanding of the goals of science. They are empirical adequacy, and more remotely practical success of applications, rather than a search for hidden reality. This may strike a realist as lowly and demeaning of science, but that again is appeal to emotions, and mechanics too once "corrupted the good of geometry", according to Plato, for it "uses bodies needing much vulgar manual labor". There is no being right or wrong about goals, they are not matters of fact. This is one reason why the dispute is perennial. Anti-realism and instrumentalism take the scientific method itself at face value, and view the ontologies it produces only as tools. Anti-realism takes an agnostic position on reality of theoretical entities, and the idealism/materialism dispute in particular, and questions if one can even make sense of "mind-independent" (as opposed to just not mind-determined) reality. Unlike realism, it is a stable position, starting at anti-realism one is anchored there, whereas starting at realism one has to resist being led away from it. And it has as much faith in science as does realism, but on its own terms.
Here is Quine's description of his faith in science in On What There Is, that an anti-realist can largely endorse:"The physical conceptual scheme simplifies our account of experience because of the way myriad scattered sense events come to be associated with single so-called objects; still there is no likelihood that each sentence about physical objects can actually be translated, however deviously and complexly, into the phenomenalistic language... Viewed from within the phenomenalistic conceptual scheme, the ontologies of physical objects and mathematical objects are myths. The quality of myth, however, is relative; relative, in this case, to the epistemological point of view. This point of view is one among various, corresponding to one among our various interests and purposes". Technically Quine self-identifies as a realist, see however How does Quine answer the metaphysician's charge that scientism is self-refuting? for the nature of his "realism".