In what sense is the world “real”? What are we asking, when we ask that question?
I don’t know. But G. Polya recommends that when facing a difficult problem, one look for similar but easier problems that one can solve as warm-ups. I would like to do one of those warm-ups today; I would like to ask what disguised empirical question scientists were asking were asking in 1860, when they debated (fiercely!) whether atoms were real.
Let’s start by looking at the data that swayed these, and similar, scientists.
Atomic theory: By 1860, it was clear that atomic theory was a useful pedagogical device. Atomic theory helped chemists describe several regularities:
- The law of definite proportions (chemicals combining to form a given compound always combine in a fixed ratio)
- The law of multiple proportions (the ratios in which chemicals combine when forming distinct compounds, such as carbon dioxide and carbon monoxide, form simple integer ratios; this holds for many different compounds, including complicated organic compounds).
- If fixed volumes of distinct gases are isolated, at a fixed temperature and pressure, their masses form these same ratios.
Despite this usefulness, there was considerable debate as to whether atoms were “real” or were merely a useful pedagogical device. Some argued that substances might simply prefer to combine in certain ratios and that such empirical regularities were all there was to atomic theory; it was needless to additionally suppose that matter came in small unbreakable units.
Today we have an integrated picture of physics and chemistry, in which atoms have a particular known size, are made of known sets of subatomic particles, and generally fit into a total picture in which the amount of data far exceeds the number of postulated details atoms include. And today, nobody suggests that atoms are not "real", and are "merely useful predictive devices".
Copernican astronomy: By the mid sixteen century, it was clear to the astronomers at the University of Wittenburg that Copernicus’s model was useful. It was easier to use, and more theoretically elegant, than Ptolemaic epicycles. However, they did not take Copernicus’s theory to be “true”, and most of them ignored the claim that the Earth orbits the Sun.
Later, after Galileo and Kepler, Copernicus’s claims about the real constituents of the solar system were taken more seriously. This new debate invoked a wider set of issues, besides the motions of the planets across the sky. Scholars now argued about Copernicus’s compatibility with the Bible; about whether our daily experiences on Earth would be different if the Earth were in motion (a la Galileo); and about whether Copernicus’s view was more compatible with a set of physically real causes for planetary motion (a la Kepler). It was this wider set of considerations that eventually convinced scholars to believe in a heliocentric universe. 
Relativistic time-dilation: For Lorentz, “local time” was a mere predictive convenience -- a device for simplifying calculations. Einstein later argued that this local time was “real”; he did this by proposing a coherent, symmetrical total picture that included local time.
Luminiferous aether: Luminiferous ("light-bearing") aether provides an example of the reverse transition. In the 1800s, many scientists, e.g. Augustin-Jean Fresnel, thought aether was probably a real part of the physical world. They thought this because they had strong evidence that light was a wave, including as the interference of light in two-slit experiments, and all known waves were waves in something.[2.5]
But the predictions of aether theory proved non-robust. Aether not only correctly predicted that light would act as waves, but also incorrectly predicted that the Earth's motion with respect to aether should affect the perceived speed of light. That is: luminiferous aether yielded accurate predictions only in narrow contexts, and it turned out not to be "real".
Generalizing from these examples
All theories come with “reading conventions” that tell us what kinds of predictions can and cannot be made from the theory. For example, our reading conventions for maps tell us that a given map of North America can be used to predict distances between New York and Toronto, but that it should not be used to predict that Canada is uniformly pink.
If the “reading conventions” for a particular theory allow for only narrow predictive use, we call that theory a “useful predictive device” but are hesitant about concluding that its contents are “real”. Such was the state of Ptolemaic epicycles (which was used to predict the planets' locations within the sky, but not to predict, say, their brightness, or their nearness to Earth); of Copernican astronomy before Galileo (which could be used to predict planetary motions, but didn't explain why humans standing on Earth did not feel as though they were spinning), of early atomic theory, and so on. When we learn to integrate a given theory-component into a robust predictive total, we conclude the theory-component is "real".
It seems that one disguised empirical question scientists are asking, when they ask “Is X real, or just a handy predictive device?” is the question: “will I still get accurate predictions, when I use X in a less circumscribed or compartmentalized manner?” (E.g., “will I get accurate predictions, when I use atoms to predict quantized charge on tiny oil drops, instead of using atoms only to predict the ratios in which macroscopic quantities combine?".
 Of course, I’m not sure that it’s a warm-up; since I am still confused about the larger problem, I don't know which paths will help. But that’s how it is with warm-ups; you find all the related-looking easier problems you can find, and hope for the best.
 I’m stealing this from Robert Westman’s book “The Melanchthon Circle, Rheticus, and the Wittenberg Interpretation of the Copernican Theory”. But you can check the facts more easily in the Stanford Encyclopedia of Philosophy.
[2.5] Manfred asks that I note that Lorentz's local time made sense to Lorentz partly because he believed an aether that could be used to define absolute time. I unfortunately haven't read or don't recall the primary texts well enough to add good interpretation here (although I read many of the primary texts in a history of science course once), but Wikipedia has some good info on the subject.
 This is a standard example, taken from Philip Kitcher.
 This conclusion is not original, but I can't remember who I stole it from. It may have been Steve Rayhawk.
 Thus, to extend this conjecturally toward our original question: when someone asks "Is the physical world 'real'?" they may, in part, be asking whether their predictive models of the physical world will give accurate predictions in a very robust manner, or whether they are merely local approximations. The latter would hold if e.g. the person: is a brain in a vat; is dreaming; or is being simulated and can potentially be affected by entities outside the simulation.