Is spacetime real?

This question leads us quite quickly to the metaphysical question of what it is that distinguishes physics from pure mathematics. That question is quite subtle so I do not think you will find a complete consensus, among experts, on the answer to your question.

To understand physics generally, it becomes more and more useful, as you progress to more subtle areas, to make clear in your mind the distinction between the mathematical abstractions, such as number, vector, tensor, Hilbert space, quantum amplitude, and the physical things such as bricks, cats, light pulses, planets, electrons, and their physical properties such as mass, charge, velocity, acceleration, and so on. When we write a symbol such as $\bf v$ and say "it is the velocity" then on the one hand we are referring to a mathematical abstraction---a vector that lives in an abstract space of mathematical abstractions---and on the other hand we are referring to a physical quantity: the rate at which an object is increasing its displacement relative to other objects. In Newtonian physics you can get away with being a bit hazy on this distinction, but in quantum physics you cannot. A quantum operator such as $\hat{\bf p}$ is not "momentum", it is "the momentum operator" or "the operator representing momentum". It is a mathematical abstraction. The physical quantity here is the amount of "oomph" an object may have owing to its mass and motion. That is not an operator but a property of the physical world, one which can be detected by physical apparatus.

In General Relativity there are reasons to say that spacetime is purely an abstraction, placing it at the mathematical end of the distinction, and also reasons to say it is somewhat physical. It is somewhat physical because, after all, it has measurable properties such as curvature, and it can transmit energy and momentum from one place to another. But its physical nature is nevertheless subtle because you cannot tell whether you are moving relative to spacetime, nor even if you are accelerating relative to spacetime---those very phrases do not have any meaning, as it turns out; they amount to a misuse of terminology. And spacetime is not a thing in the sense of "made of stuff which has energy" because the energy we associate with spacetime (the energy which appears and disappears when gravitational radiation is emitted and absorbed) does not contribute to the stress-energy tensor in the Einstein field equation. This makes the role of spacetime unique among the sorts of phenomena we study in physics.

Another way to argue that spacetime is physical, not merely a mathematical tool, is that you can study many differentiable manifolds using pure maths, but only one of them will be able to give a precise description of physical goings-on as observed in the universe. What singles out this manifold from all the others? If you take the view that spacetime is purely a mathematical tool then it is hard to answer---logically, the special manifold would have to have something further about it that shows that it has this special role, but pure maths is unable to determine that special further property. We can avoid this puzzle by asserting that there is a physical "thing"---i.e. physical spacetime---and the special manifold is the one which correctly describes the nature of this physical "thing".

In view of all this, I for one feel that it is more helpful than not to say that spacetime is indeed a physical "thing", as long as one keeps in mind the various caveats which people who take the other view will also wish to emphasize.

You can ask this question of most anything we study in physics. Regardless of what you are asking about you can usually get different answers from different physicists. The reason is that “real” is primarily a philosophical concept (in metaphysics) rather than a scientific concept. We don’t have a real-o-meter that we can use to quantity the reality of something.

Spacetime is indeed a mathematical construct (a pseudo-Riemannian manifold), but there is also a very clear mapping of that mathematical construct to experimental measurements of length, duration, angles, etc. Spacetime is a form of geometry, but all experiments indicate that it is the geometry of nature.

The mathematical construct of spacetime is isomorphic to experimental measurements of distance, direction, and duration, so it is typically not problematic to use the same term to refer to the mathematical construct and also to the physical measurements. But one place where it can become problematic is if one person is using it to refer only to the mathematical concept and another is using it to refer only to the physics and another is using both meanings interchangeably. I believe that is what has happened in the linked answer.

I don’t think it is necessary to take a stance on the reality of spacetime. Just be aware that the same word is used to refer to two separate but related concepts. If you ever come into a confusing conflict, it is likely just a discrepancy on which of the two related concepts are being discussed.

Something which must be understood if one wants to correctly interpret physics as a subject is the relationship between mathematical models and the real world. The entire purpose of physics is to make measurements in the real world, identify patterns, and then search for similar patterns that exist in mathematics. Once we have found said mathematical patterns we can then use these to make predictions about the real world.

It is a common mistake (and unfortunately one that is often propagated by popular science outlets) to believe that the mathematical structures that we use to make predictions about the real world really exist in the real world.

Classical electrodynamics for example models the electric and magnetic fields as vector fields, but this does not imply that there exist tiny arrows at every point in space that represent the electric and magnetic fields.

General relativity (GR) is another example (and probably the most frequent offender of this kind). Mathematically, GR is an applied theory of differential geometry, which involves (very loosely) studying the properties of surfaces. The four-dimensional surface used in GR is called spacetime, the curvature of this surface is used to make real-life predictions about the behaviour of gravity.

However, just as classical electrodynamics does not imply that tiny arrows exist at every point in space, GR does not imply that there is a real-life surface on which we live that curves in the presence of mass/energy. A more accurate way of putting this would be that physics does not predict what nature is, only how it works.

So rather than suggesting that the real world somehow "contains" mathematics, which implies that mathematics is special and has intrinsic meaning in nature, it is perhaps more accurate to simply note that we appear to live in a world that has immense, rigid structure and consistency, both of which are build-in features of mathematics.