Are there pure sine waves in nature or are they a mathematical construct that helps us understand more complex phenomena?

Since no phenomenon is completely periodic (nothing keeps repeating from minus infinity to infinity), you could say that sine waves never occur in nature. Still, they are a good approximation in many cases and that is usually enough to consider something physical.

Or are they a mathematical construct that helps us interpret nature?

I would even go further and say that it is reasonable that everything in physics is a mathematical construct that helps us interpret nature, but that would lead to the philosophical debate of what is nature an so on. After all, almost everything in physics breaks down or at least becomes problematic at some regime: the notion of particles in strongly-interacting theories, energy in general relativity, the notion of a sound wave at the atomic scale...

This is really more of a supplement to jinawee's answer, but you might want to consider what, if anything, makes your question different from the following analogous questions:

• Are there lines in nature, or are they a mathematical construct that helps us understand more complex phenomena?
• Are there points in nature, or are they a mathematical construct that helps us understand more complex phenomena?
• Are there spheres in nature, or are they a mathematical construct that helps us understand more complex phenomena?

At a fundamental level, physics is about building mathematical models of the observable world. These models are "real" only to the extent that they make testable predictions that can be checked against that observable world. Since any experimental observation is only accurate up to a certain precision, it's never possible to say that one of these mathematical models is exactly the same as the thing that it describes. But without the language of mathematical idealizations, physics would be unable to do much of anything.

As jinawee said they cannot be physical due to their temporal extent. Nevertheless they are extremely useful because they (sine, cosine and combination of them) are the eigen-functions of the operator $\partial_t^2$ that shows up in a lot of differential equations:

$\partial_t^2(A\sin(\omega t+\phi))=-\omega^2A\sin(\omega t+\phi)$.

You can easily check that this is also true for any linear combination of sine and cosine of the same frequency with arbitrary constant phase. On the other hand, this property (eigen-function) is not satisfied by other functions, even periodic ones like $\sin(\omega t)+\sin(2\omega t)$. This why these functions are special and omnipresent.

Now since wave-like equations are often linear we are naturally lead to use Fourier analysis: we can decompose ~any signals* as a linear combination of harmonic function sine and cosine, transform the derivative operators into algebraic multiplication $(\partial_t^2 \rightarrow -\omega^2)$ easily solve the now algebraic equation, and add together the solutions to recreate a physical signal (i.e. bounded in time). Check Wave- vs Helmholtz- equation

*there are some restrictions but this is not bothering for usual physical signals