We consider $n$-tuples of commuting operators $a=a_1,...,a_n$ on a Banach space with real spectra. The holomorphic functional calculus for $a$ is extended to algebras of ultra-differentiable functions on ${ℝ}^n$, depending on the growth of $\parallel exp(ia·t)\parallel$, $t\in {ℝ}^n$, when $\left|t\right|\to \infty$. In the non-quasi-analytic case we use the usual Fourier transform, whereas for the quasi-analytic case we introduce a variant of the FBI transform, adapted to ultradifferentiable classes.