If a fundamental theory exibits e.g. a mirror symmetry, in what sense it the underlying geometry real? 
Are the more recently discovered symmetries in string theory such that the theories based on mirroring geometries are absolutely the same from an observable point of view?

I have mirror symmetry in mind, i.e. by symmetries I mean the dualities encountered in string theory.

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Say you take e.g. take the Schwarzschild metric from general relativity and consider its quivalent in some theory of quantum gravity theory. There, if you don't look close enough, i.e. if you look at averages, the quantum-Schwarzschild space would look classical, namely like the general relativity Schwarzschild space. 
Now the target space in string theory is one manifold and it leads to a quantized theory. But if there are now actually two mirroring manifolds where this theory can come from, the I guess the classical version looks like neighter of these geometries. The following question comes from this observation, that there are totally different (mirroring) options with which Calabi-Yau you can start (considered in the Lagrangian) if you go in the quantization direction and the conclusion that neighter of will be the limit, or there are more limits. 

If we assume that the world is described by such a theory, do we besically live in two indestingushable geometries at once?

To put it differently, I ask if string theory can be considered a quantum gravity in the sense that there is a classical limit down to classical differential geometry, especially regarding the extra dimensions. And if so, to what geometry in these higher dimensions does the limit lead? I do know that at least some of the geometry must come from the string excitation representing a quantizations of a metric/graviton. But I suspect different metrics in the non-linear sigma model Lagrangian will also affect the classical picture in some essential way, directly or not.
 A: For two manifolds related by a mirror symmetry, all the predictions are the same if it is IIA strings on one vs. IIB strings on the other--- the two ideas, compactify IIA on M or compactify IIB on mirror-of-M are identical, and the two manifold/string-theory pairs may not be distinguished in any way--- there is no difference between them, they are the exact same theory in two different languages. It is not meaningful to ask which is right.
But IIA theory on M is not the same as IIA theory on the mirror. The easiest way to understand this is using a circle, where mirror symmetry is T-duality, and the mirror symmetry is just the generalization of this to the general Calabi Yau which is mathematically most interesting, because it tells you that the IIA string spectrum on one manifold is equal to the IIB string spectrum on the other.
The identity of T-duality (known already to Schwarz and collaborators in the 1980s) means that there is no difference really between IIA and IIB theory, they are the same theory with a different geometrical language for the microscopic degrees of freedom. The duality is fascinating, but whenever one geometry becomes large and classical (decompactified), the dual description becomes remote and super-quantum (sub-planckian). So generally, you know which description you should be using.
