Does overall phase matter? If it's experimentally known that a measurement of $S_x$, after measuring $S_z$, results in $\hbar/2$ half of the times, and $-\hbar/2$ the other half, is it correct to claim that
$$|z\pm\rangle =\frac{1}{\sqrt2}\left( |x+\rangle \pm e^{i\delta_1}|x-\rangle\right), \tag{1}$$
without caring about the overall phase? If yes (which, by the way, Sakurai claims to be correct in his book), then it similarly follows that 
$$|z\pm\rangle =\frac{1}{\sqrt2}\left( |y+\rangle \pm e^{i\delta_2}|y-\rangle\right). \tag{2}$$
What's not so fine is that solving for $|x+\rangle$ and $|y+\rangle$ gives us
$$|x+\rangle = \frac{1}{\sqrt2}\left( |z+\rangle + |z-\rangle\right) = |y+\rangle \tag{3}$$
which is obviously wrong. $|x+\rangle$ cannot be equal to $|y+\rangle$.
So where did things go wrong? Does it all lie in ignoring the overall phase?
 A: You can only ignore the overall phase of a ket when you evaluate a physical quantity related to it. You can't dump overall phases everywhere when you're doing math on the kets themselves.
For example, suppose we have two kets $|x \rangle$ and $|y \rangle$ and we define
$$|z \rangle = |x \rangle + |y \rangle.$$
By your logic, this equation remains true under any phase redefinition of $|x \rangle$ and $|y \rangle$, so we have
$$|z \rangle = e^{i\theta_1} |x \rangle + e^{i\theta_2} |y \rangle$$
This is only true if the two $|z \rangle$'s are the same, i.e. if $\theta_1 = \theta_2$, in which case $|z \rangle$ itself just picks up an overall phase. If this isn't true, then you've mistakenly thrown away a relative phase between the $|x \rangle$ and $|y \rangle$ components, which has physical consequences.
Another way of saying this is that 'global phase' is only a single degree of freedom. You can rotate away any one phase in a problem, but you can't do that to all of them.
A: In this case the measurement given (projection along z) just doesn't tell you what the phase is, it doesn't mean that the phase is inconsequential. I.e., this means that any state of the form
$$|\psi\rangle = \frac{1}{\sqrt{2}}(|\uparrow\rangle+exp\{i \phi\} |\downarrow\rangle)$$
Yields the given probabilities. The error is then in the equals sign in equation 2. Just because the measurement outcomes are the same doesn't mean the states are equal. 
Incidentally this poses a general problem if you want to completely know your quantum state by doing measurements on it. This principle implies that you must necessarily take measurements in more than one basis in order to fully define your state. This is heavily explored and studied in the field of quantum state tomography.
