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