When you do a measurement, the information of the phase of the resulting projected eigenstate with the former full state is lost to you, but it is still available to other observers.
This is because of the nature of entanglement; when two systems initially unentangled interact, assuming one of them is largely classical and the other is quantum (in superposition of multiple non-degenerate states), the final state is largely described by a sum of entangled pairs, of quantum eigenstate $\times$ a classical system coupled to that eigenstate. This is the Everett picture of the measurement.
What eigenstate basis? it depends on the hamiltonian of the measurement apparatus, different apparatus will couple to the quantum system differently, making the entanglement along different basis.
If you think on the sums of the resulting state after measurement, the i-th classical observer that sees the i-th eigenstate, did lose all phase information, and worse any physical way to retrieve it back, even in principle. After a quantum system has been projected to your apparatus, there is nothing you can do to retrieve the non-projected component of the wavefunction. Measurements are inherently non-unitary steps of a system evolution.
But this is not agreed by all observers; other observers (let's call them Meta) that have not interacted directly with either the system or the observer (described by the classical component) will still see both systems to interact and evolve unitarily at all times; if he is clever enough and make the appropiate measurements he can confirm that the measurements done by the classic observer has not erased any phase information, but for all practical purposes, the resulting phase of the coupled pair is a complex result of both systems interaction and initial phases, so it would in general average out as random noise.
So in short, unitarity as a history of the evolution of systems is not an universal property agreed by all observers, it will depend on what systems they are entangled and what measurements (intentional or not) have they done. Note however, that this is not the same as "subjectivity", but more akin to how different observers in relativity might see different quantities like elapsed time, curvature, etc.