Suppose that we have a nondegenerate quantum system with an orthonormal basis of eigenstates $|\psi_n\rangle$ with eigenvalues $E_n$. Consider a wavefunction $\psi(x,t)=\sum_n c_n|\psi_n(x,t)\rangle$. Suppose that we measure the energy at time $T$ and obtain value $E_m$. Now the system collapses to the eigenstate $|\psi_m\rangle$. However, what will be the new phase factor? The new wavefunction must be equal to $$A|\psi_m\rangle$$ for some complex number $A$ with $|A|=1$. Is there a rule to determine the value of $A$? With a `rule', I mean a mathematical formula with the numbers $c_n$ and the physical properties of the system as input and $A$ as output.
If the measurement measures phase, e.g., using a kind of Aharonov-Bohm device, then the phase constitutes a quantum number in the spectrum of the measurement operator - that is, in the process of measurement we project the wave function on a state with a specific phase (usually known up to an additive constant).
Otherwise the phase is not relevant (not determined), as it will not influence the results of measurement.