  Given a separable Banach space $E$, we construct an extremely non-complex
Banach space (i.e. a space satisfying that $\|Id + T^2\|=1+\|T^2\|$ for every
bounded linear operator $T$ on it) whose dual contains $E^*$ as an $L$-summand.
We also study surjective isometries on extremely non-complex Banach spaces and
construct an example of a real Banach space whose group of surjective
isometries reduces to $\pm Id$, but the group of surjective isometries of its
dual contains the group of isometries of a separable infinite-dimensional
Hilbert space as a subgroup.
