  We show that any irreducible representation $\rho$ of a finite group $G$ of
exponent $n$, realisable over $\mathbb{R}$, is realisable over the field
$E:=\mathbb{Q}(\zeta_n)\cap\mathbb{R}$ of real cyclotomic numbers of order $n$,
and describe an algorithmic procedure transforming a realisation of $\rho$ over
$\mathbb{Q}(\zeta_n)$ to one over $E$.
