  It is well known that the Mach-Zender interferometer exhibits a trade-off
between the a priori which-path knowledge and the visibility of its
interference pattern. This trade-off is expressed by the inequality
$\mathcal{P}^2 + \mathcal{V}^2 \leq 1$, constraining the predictability
$\mathcal{P}$ and visibility $\mathcal{V}$ of the interferometer. In this paper
we extend the Mach-Zender scheme to a setup where the central phase shifter is
substituted by a generic unitary operator. We find that the sum $\mathcal{P}^2
+ \mathcal{V}^2$ is in general no longer upper bounded by $1$, and that there
exists a whole class of interferometers such that the full fringe visibility
and the full which-way information are not mutually exclusive. We show that
$\mathcal{P}^2 + \mathcal{V}^2 \leq L_U$, with $1 \leq L_U \leq 2$, and we
illustrate how the tight bound $L_U$ depends on the choice of the unitary
operation $U$ replacing the central phase shifter.
