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Gravitational waves carry energy, momentum, and angular momentum from one point to another. In my book, this makes spacetime just as real as an electromagnetic field, which also transports these quantities. You feel electromagnetic energy whenever you walk outside on a sunny day. If gravity weren’t so weak, you would feel the energy in gravitational waves.

Most physicists consider these fields just as real as matter, which in modern physics is
explained using yet more fields!

spacetime is three dimensions of space and one of time. ... my question is, whether spacetime is real or is just a mathematical construct

Here is different from there and now is different from then whether you do the math or not. A mathematical construct cannot be the difference between a near miss and a catastrophic collision.

Spacetime furnishes the stage upon which all the physics in the universe is acted out, in one way or another. For it to behave in the way we have observed it to, it must have certain properties, and our measurements of those properties are in close correspondence with calculations based on our mathematical models.

Based on this, spacetime is not just a mathematical construct invented to make calculations easier. The constructs are our mathematical models, which are intended to represent the functioning of the universe as it is.

Several problems with the question. What is real? What is time? What is space?

If real is reality to humans; space being the 3 dimensional world metastructure; time being the mapping of change to space giving a 4th dimension. If these are the case, I believe we can answer yes to your question, spacetime WOULD BE real. The problem is that the question itself has many hypothetical predications. Leading to the answer inheriting the curse (the curse of ifs?).

I don't think physics will ever gives us absolute answers that satisfy this direction of questioning as it has humans in the centre of the question making it subjective. Reality, whatever it is, is not the reality people are used to in their normal lives. In science, the reality is made of models that have a logical structure through maths, and human languages. Our outlook on the world limits our understanding of it leading to cases in which theories have to be made. Theories are great, but not absolute. This leads to science being incomplete (possible forever).

I would argue that theories that contradict one and another lead to the existance of many possible realities in science. Some are more propable than others given context, sizing of the system, and time of relevance. A theory of everything would unify these realities, and maybe give an answer to your question. I hope we find one.

In philosophy of physics, there are two main positions on the reality of spacetime: substantivalist and relationalist. The former sees spacetime as more of a real "substance", the latter sees spacetime as only emergent and not an underlying reality. Other answers have alluded to these positions, but not labelled them explicitly.

Some thinkers love Einstein's 1916 quote:

Nothing is physically real but the totality of space-time point coincidences.

Such "coincidences" refer to the point of crossing of two worldlines, for instance. I hear this quoted by those who seek to incorporate Mach's principle more fully into general relativity (or alternate theories). I have also seen it advocated in quantum gravity.

In the least, I hope this answer provides a few terms to look up for further information.

Spacetime is wholly a mathematical construct and not a real thing since it highly depends on the observer-learner qualities.

Spacetime provides a language or vocabulary that allows to us talk in a very precise way about the motion of an idealised point object. The motion is represented by a curve or "worldline" in spacetime, each point on the curve has four real co-ordinates, and the curve as a whole can be represented by parametric equations that tell us how these co-ordinates are related at different points on the curve. All of this language is imported from the mathematical field of geometry, and specifically the geometry of differential manifolds.

Using the same language we can define a notion of distance along a curve (a "metric") and with a given metric we can distinguish particular curves called "geodesics" which generalize the notion of a straight line. The interesting thing is that if we make the right choice of geodesic then the distance along a worldline correspond to the elapsed time that passes for an object following that worldline, and the worldline of an object in free fall (with no external forces acting on it apart from gravity) is a geodesic.

The Einstein field equations then show us how to choose the "right" metric - they are a set of non-linear differential equations that tell us how this metric changes as we move across the spacetime manifold. The physical input that "drives" these differential equations is the distribution of matter and radiation (i.e. the electromagnetic field) across the spacetime manifold.

So the spacetime manifold is a mathematical construct, but it is one that allows us to talk about how real physical objects move along their worldlines and interact with (i.e. influence the worldlines of) other objects - and to do this in a very precise, quantitative way.

I can interpret your question in two different ways.

1. "Is spacetime real?" <=> "Do real triangles exist"?

This is a ~2.5k years old idea which you can read about in more depth in Platonic Realism or the primary sources linked from it. One example from that page is that while you can certainly take a pencil, draw a triangle on a piece of paper, call that a triangle, and convince yourself that triangles exist in the real world (because you just created one), there is something completely different also: An ideal triangle, or in other words, "a polygon with three edges and three vertices". And this is not real (if you subscribe to Platonic Realism). Simply speaking, nowhere in the universe you will ever find three edges connecting three vertices. Those objects are all just ideas in our brains, abstract concepts, fantasy, or whatever you want to call them.

In this sense, our mathematical construct labeled spacetime is not real. It is just that, a mathematical construct, which you could, if you wanted to, trace back to a handful of axioms. It just so happens that if you go and make some calculations (which, also, are of course not "real"; you clearly do not influence the universe in any way by pushing your numbers around), the mathematical construct of spacetime seems to somehow come out to match what we are measuring in the real world. That's the nature of physics - it tries to explain observations; it retrofits mathematical constructs to the real world. But never the other way around. Nobody in their right mind would suppose that the real world behaves like it does because of our mathematical construct of spacetime.

And it would not matter either way. Our physics do not define reality; we can be happy if we are able to speak about reality with our physics.

2. "Is spacetime real?" <=> "Is there 'ether'?"

If you are asking whether there is some substance, or "stuff" which is spacetime in reality, then the answer is also no. Yes, people may call it a "fabric", but that is just a word. It is not like a "canvas" or "piece of cloth" where the real physical objects are kind of painted on. The problem might be that we need some didactic tools to bring basic concepts over to the audience; we are all familiar with pictures of curved spacetime, where a planet sits in a bowl-shaped indent in some semi-2D spacetime picture. This is complete nonsense, physically. Sure, you can put layer upon layer of abstraction on the actual maths, and end up with something like that; or you can just explain it like this to make your reader/viewer feel that they know something about how the universe works. But it is not that.

This is the same as asking "what is an electron really". Nobody knows. We only know what our physical formulae / maths tell us about that phenomenon we call "electron", but we do not (and arguably cannot ever) know what it actually is. The same goes for basically everything else. We can always just approximate, and everything is filtered through our senses - in this kind our "brain sense" handling the mathematics of spacetime. We're still in the cave, and there is no reason to believe that we will ever come out.

N.B., I find it fascinating that Plato figured all of that out 2.5 kilo-years ago, and that mankind manages to keep remembering that until this day. I'm pretty sure little of what we're figuring out today will be around in 2500 years from now...

Well, if you can imagine a real difference in the consequences between doing something right now or doing it next week, spacetime must be as real as just observeable space. Cause and effect is a space time thing, and last I checked we depend on it in all we do.

For some years I have been puzzled by the concept of the “twin paradox”; consider the following.
Imagine you are in a space ship travelling at 0.75 times the speed of light and its cabin is 4 units of distance wide and 3 units of length long. A photon traversing the width of the cabin takes 4 units of length divided by the speed of light for the transit. A fixed observer sees the photon traveling a distance equal to the square root of 4 units of length squared plus 0.75 times 4 units of length squared i.e. 5 units of length. Hence a time dilation of 5 divided by 4, 1.25
Now let us fire a photo from the forward corner of the cabin to the aft corner of on the opposite side. A distance of 5 units of length. The fixed observer sees the photon travel only 4 units of distance because at the time the photon arrives the aft bulkhead has move forward 3 units of distance. This suggests a time dilation of 4 dived by 5, 0.8
This cannot be correct, is there a fault in the theory! Please have one of your experts clarify for me.