  We investigate the possibility of constructing exponentially localized
composite Wannier bases, or equivalently smooth periodic Bloch frames, for
3-dimensional time-reversal symmetric topological insulators, both of bosonic
and of fermionic type, so that the bases in question are also compatible with
time-reversal symmetry. This problem is translated in the study, of independent
interest, of homotopy classes of continuous, periodic, and time-reversal
symmetric families of unitary matrices. We identify three $\mathbb{Z}_2$-valued
complete invariants for these homotopy classes. When these invariants vanish,
we provide an algorithm which constructs a "multi-step" logarithm that is
employed to continuously deform the given family into a constant one,
identically equal to the identity matrix. This algorithm leads to a
constructive procedure to produce the composite Wannier bases mentioned above.
