To set a context for my understanding of your question let's start with a single qubit. Then, the state can be described by two complex numbers $|\psi\rangle=a|0\rangle+b|1\rangle$ subject to a constraint $|a|^2+|b|^2=1$ and up to an equivalence $|\psi\rangle \sim e^{i\phi} |\psi\rangle$. Absolute values of $a$ and $b$ determine probabilities to obtain either state $|0\rangle$ or $|1\rangle$ when measured in the computational basis ($|a|^2$ and $|b|^2$, respectively). These probabilities are insensitive to phases of $a$ and $b$. However, for other measurements the relative phase between $a$ and $b$ does matter. For example, states $|\pm\rangle=\frac{1}{\sqrt{2}}\left(|0\rangle\pm |1\rangle\right)$ are orthogonal and have different probabilities when measured in the basis $|\pm\rangle$.
For two qubits there will be four coefficientn $|\psi\rangle = a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle$ subject to $|a|^2+|b|^2+|c|^2+|d|^2=1$ and up to a global phase $|\psi\rangle\sim e^{i\phi}|\psi\rangle$. Again, for measurments in the computational basis only absolute values of $a,b,c,d$ are relevant. However, for any generic measurement phases of these complex numbers are also essential. For multi-qubit states you can assign phases to each vector in the computational basis (the phase of the corresponding coefficient) but that is not the same as phases of individual qubits. In fact, I do not think there any sense to talk about phases of individual qubits when they are part of the multi-qubit system.
Your example with the GHZ state is a special case where you only consider two basis vectors (among 8 possible for a 3-qubit state) and call a relative phase between them your "one phase". A general 3-qubit state would be described by 8-1=7 phases (and 8-1=7 probabilities in the computational basis).
If I understand correctly, you assume that in a multi-qubit state individual qubits still must have some "phases". This is incorrect, as individual qubits do not even have (pure) states! In sharp contrast to classical mechanics a state of a composite system in quantum mechanics can not generally be described by describing each part of the system individually. This is only true for unentangled states. If a qubit is entangled with others it generally can only be described by a density matrix. In your example, the GHZ state is maximally entangled. The density matrix of any individual qubit is $\rho=\begin{pmatrix}\frac12&0\\0&\frac12\end{pmatrix}$ and it is as far from any pure state as you can get. Operationally, measuring such qubit in any basis gives completely random results (for pure states there is always a basis with definite results of the measurements).
