I consider the problem of efficiently solving a system of n non-linear equations in R^d, where each equation is random homogeneous polynomial of arbitrary degrees. In the complex case and for n=d−1, Beltran and Pardo proved the existence of an efficient randomized algorithm and Lairez recently showed it can be de-randomized to produce a deterministic efficient algorithm. Here we consider the real setting, to which previously developed methods do not apply. We describe an algorithm that efficiently finds solutions (with high probability) for n=d−O(sqrt{dlog d}). [Based on joint work with Eliran Subag]