  We consider a real periodic Schr\"odinger operator and a physically relevant
family of $m \geq 1$ Bloch bands, separated by a gap from the rest of the
spectrum, and we investigate the localization properties of the corresponding
composite Wannier functions. To this aim, we show that in dimension $d \leq 3$
there exists a global frame consisting of smooth quasi-Bloch functions which
are both periodic and time-reversal symmetric. Aiming to applications in
computational physics, we provide a constructive algorithm to obtain such a
Bloch frame. The construction yields the existence of a basis of composite
Wannier functions which are real-valued and almost-exponentially localized.
  The proof of the main result exploits only the fundamental symmetries of the
projector on the relevant bands, allowing applications, beyond the model
specified above, to a broad range of gapped periodic quantum systems with a
time-reversal symmetry of bosonic type.
