Can you give an experimental example showing the difference between global and relative phase in QM? Let's say, that we are in possesion of a very simple quantum system, whose state can be written as
$$ |\psi\rangle = c_0 |\psi_0 \rangle + c_1 |\psi_1\rangle.$$
Now, we can change this state in two ways:
1) By adding global phase:
$$ |\psi'\rangle = e^{i\theta}|\psi\rangle  = e^{i\theta} (c_0 |\psi_0 \rangle + c_1 |\psi_1\rangle).$$
2) By adding a relative phase difference between $|\psi_0\rangle$ and $|\psi_1\rangle$:
$$ |\psi''\rangle = c_0 |\psi_0 \rangle + e^{i\phi}c_1 |\psi_1\rangle.$$
I know that in an experiment we cannot detect the global phase from case 1), but we can measure the relative phase from case 2). But I cannot figure out any tangible example of these phases. Can you give one?
(*) Additional question: I've heard something about the Berry phase. Can you explain how this one relates to the first two? If you are able to even give some "real world" example showing this relation, it would be wonderful!
 A: You don't even need quantum mechanics to measure the relative phase. You just need a classical optics, for example the 2-slit interference pattern.
The two-slit ($1$ and $2$) interference pattern gives you an electric field $E = E_1 + E_2$, whose intensity $I \propto  |E|^2$ is shown in red below:

By placing a phaseplate on slit $2$, the field from that slit becomes $E_2 \mathrm{e}^{\mathrm{i}\phi}$ so that the total field  is $E = E_1 + E_2\mathrm{e}^{\mathrm{i}\phi}$. In the example below, I chose $\phi = \pi$ so that all the maxima turn into minima and vice versa. 

To jump to quantum, you can think electrons (instead of light) incident on the two slits. Instead of the electric field, you have the probability amplitude $|\psi_1\rangle$ and $|\psi_2\rangle$. Same maths.

For the Berry phase, that's the generalisation of a geometric phase. This is a phase factor that arises from the geometry/topology of the real or Hilbert space.
The easiest way to visualise it is with parallel transport (gif from here). Moving a vector along a curved surface makes it build up an angle. This angle would be related to the Berry phase. The curved surface could be the Hilbert space of a quantum system. The vector could be a spin-1/2 atom atom.

