  In the past 20 years, the study of real eigenvalues of non-symmetric real
random matrices has seen important progress. Notwithstanding, central questions
still remain open, such as the characterization of their asymptotic statistics
and the universality thereof. In this letter we show that for a wide class of
matrices, the number $k_n$ of real eigenvalues of a matrix of size $n$ is
asymptotically Gaussian with mean $\bar k_n=\mathcal{O}(\sqrt{n})$ and variance
$\bar k_n(2-\sqrt{2})$. Moreover, we show that the limit distribution of real
eigenvalues undergoes a transition between bimodal for $k_n=o(\sqrt{n})$ to
unimodal for $k_n=\mathcal{O}(n)$, with a uniform distribution at the
transition. We predict theoretically these behaviours in the Ginibre ensemble
using a log-gas approach, and show numerically that they hold for a wide range
of random matrices with independent entries beyond the universality class of
the circular law.
