Suppose I have a wave-function over a Hilbert-space of (complex) dimension $N$. It has $2 N-2$ real degrees of freedom, after normalization and removing the phase. It seems to me that I can measure these degrees of freedom with $2N-2$ measurements, first by projecting on each of the basis states, and then by making a measurement for each of the relative phases.
Lucien Hardy in his 2001 paper about the "reasonable axioms" (https://arxiv.org/abs/quant-ph/0101012) says it takes instead $N^2 -2$ measurements to completely determine the state, by which he means the density matrix (I am referring to normalized pure states, otherwise it's $N^2$). He calls these "fiducial" measurments.
I understand where the $N^2$ comes from -- it's the number of real entries that you need to specify a generic hermitian matrix over a complex vector space of dimension $N$.
What I don't understand is why do I need $N^2-2$ measurements if I know that the density matrix of a pure state can be written as a tensor product of the wave-function and therefore has only $2N-2$ degrees of freedom? Why do the measurements that I mentioned above (projection on basis plus relative angles) not entirely determine the density matrix? Or do I misunderstand what the "fiducial" measurements are?