  We study dynamics of a single qubit encoded in two pairs of Majorana modes,
whereby each pair is hosted on a trijunction described by the Kitaev model
extended by many-body interactions. We demonstrated that the challenging
phase-gate may be efficiently implemented via braiding of partially overlapping
modes. Although such qubit acquires both geometric and dynamical phases during
the braiding protocol, the latter phase may be eliminated if the Majorana modes
are hosted by systems with appropriate particle-hole symmetry.
