  We study discrete-time quantum walks on $d$-cycles with a position and
coin-dependent phase-shift. Such a model simulates a dynamics of a quantum
particle moving on a ring with an artificial gauge field. In our case the
amplitude of the phase-shift is governed by a single discrete parameter $q$. We
solve the model analytically and observe that for prime $d$ there exists a
strong complementarity property between the eigenvectors of two quantum walk
evolution operators that act in the $2d$-dimensional Hilbert space. Namely, if
$d$ is prime the corresponding eigenvectors of the evolution operators obey
$|\langle v_q|v'_{q'} \rangle| \leq 1/\sqrt{d}$ for $q\neq q'$ and for all
$|v_q\rangle$ and $|v'_{q'}\rangle$. We also discuss dynamical consequences of
this complementarity. Finally, we show that the complementarity is still
present in the continuous version of this model, which corresponds to a
one-dimensional Dirac particle.
