  In the paper, we study the asymptotic distribution of real algebraic integers
of fixed degree as their na\"{\i}ve height tends to infinity. For an arbitrary
interval $I \subset \mathbb{R}$ and sufficiently large $Q>0$, we obtain an
asymptotic formula for the number of algebraic integers $\alpha\in I$ of fixed
degree $n$ and na\"{\i}ve height $H(\alpha)\le Q$. In particular, we show that
the real algebraic integers of degree $n$, with their height growing, tend to
be distributed like the real algebraic numbers of degree $n-1$. However, we
reveal two symmetric "plateaux", where the distribution of real algebraic
integers statistically resembles the rational integers.
