  Let L/K be a 2-birational CM-extension of a totally real 2-rational number
field. We characterize in terms of tame ramification totally real 2-extensions
K'/K such that the compositum L'= LK' is still 2-birational. In case the
2-extensions K'/K is linearly disjoint from the cyclotomic Z2-extension Kc/K,
we prove that K'/K is at most quadratic. In the other hand we construct
infinite towers of such 2-extensions.
