  It was recently proved by Bayart et al. that the complex polynomial
Bohnenblust--Hille inequality is subexponential. We show that, for real
scalars, this does no longer hold. Moreover, we show that, if
$D_{\mathbb{R},m}$ stands for the real Bohnenblust--Hille constant for
$m$-homogeneous polynomials, then
$\displaystyle\lim\sup_{m}D_{\mathbb{R},m}^{1/m}=2.$
