  Quantum systems usually travel a multitude of different paths when evolving
through time from an initial to a final state. In general, the possible paths
will depend on the future and past boundary conditions, as well as the system's
dynamics. We present a gedanken experiment where a single system apparently
follows mutually exclusive paths simultaneously, each with probability one,
depending on which measurement was performed. This experiment involves the
measurement of observables that do not correspond to Hermitian operators. Our
main result is a scheme for measuring these operators. The scheme is based on
the erasure protocol [Phys. Rev. Lett. 116, 070404 (2016), arXiv:1409.1575] and
allows a wide range of sequential measurements at both the weak and strong
limits. At the weak limit the back action of the measurement cannot be used to
account for the surprising behavior and the resulting weak values provide a
consistent yet strange account of the system's past.
