As the responses show, there are a number of quick "probably real" tests.  I'm sure that the problem is undecidable, in general, though.  For example, it's easy to see that all three of the roots of the polynomial $p(x)=13x^3-13x-1$ are real (even if they don't look it), yet they don't pass any of the test here.

    roots = x /. Solve[13 x^3 - 13 x - 1 == 0, x]
    Internal`RealValuedNumericQ /@ roots

![enter image description here][1]


  [1]: https://i.sstatic.net/KKEbT.png