The answer over the **reals** is a resounding yes: not only can you decide if the system has solutions, but you can even produce them (over other fields, not so much, e.g. $\mathbb{Q}$).

The decision problem was solved by Tarski (and later Seidenberg). Though the proof gave a theoretical way of finding solutions, it was not a practical one.

If you want to **produce** solutions, and if your system is zero-dimensional (over the complexes), then your problem is simply to determine if one of the solutions is real. First find a Grobner basis for your ideal. Through the use of a separating element in the quotient ring $\mathbb{C}[X]/I$, you can reduce to a question to determining if a certain **univariate** polynomial has real roots, which can be done with a Sturm sequence.

The positive dimensional case can be tackled by a variation on this theme, by producing a zero-dimensional system contained in your variety (by taking the critical locus of suitable functions). It's easier said than done, but software solving these kinds of problem has now been around for almost 10 years. These algorithms can be modified to handle inequalities as well as equations.

Another approach to these problems is the Cylindrical Algebraic Decomposition which is much slower, has larger output, but has the added advantage handling inequalities without needing a different algorithm.

The best reference for this is the Springer book Algorithms in Real Algebraic Geometry
by Saugata Basu, Richard Pollack, and Marie-Françoise Roy. It is freely downloadable from that link.
Nowadays, even generalist Computer Algebra Systems such as Maple or Mathematica have decent implementations of these algorithms that can breeze through your example and others of the same caliber.