Note that if you don't keep track of the phase $i$ in $Y = iXZ$, so letting $\hat{Y} = XZ$, then something 'weird' happens:
$$
\hat{Y}\hat{Y} = XZXZ = X(-XZ)Z = -XXZZ = -I.
$$
This is not just an oddity. Any stabilizer code with a generator $G_{1}$ containing an (odd number of) $Y$ will now not be possible, because:
$$
G_{1}G_{1} = -I \in \mathcal{S}.
$$
But this is of course a contradiction, because $-I$ has no $+1$ eigenstates, and therefore cannot be part of the stabilizer.
By getting rid of that $i$ explicitly you thus destroy the structure of the code.
This is not to say that you explicitly need to keep track of the phase; many people use the binary formalism (see section 'Relation between Pauli group and binary vectors'). Here you also don't track the phase.
You are looking for the normalizer of the real Pauli group in the real Unitary group, which is the Orthogonal group. Because the commutation relations of the Paulis need to be preserved under conjugation of any element of the normalizer, all that such an operations can do is permute the elements $\{\pm 1\}\otimes\{X,\hat{Y},Z\}$. Table $1$ on page $20$ of this paper lists all such permutations (there are $24$, so that's why there are $24$ elements in the single-qubit Clifford group). The authors also decompose the permutations into different generators for the Clifford group, namely two half-rotations along either the $X-,Y-$ or $Z$-axes. I really think those generators are the most intuitive; you can also relatively easily adapt them to 'take out' the imaginary parts by distilling a global phase.
As noted in the other answer, another set of generators that you can use is $\{H,Z\}$, and equally $\{H,X\}$.
However, I would personally use the elementary permutations $\{1,2,3\} \rightarrow \{1,-3,2\}$ and $\{1,2,3\} \rightarrow \{-2,1,3\}$ (with $\{1,2,3\}$ indicating $\{X,Y,Z\}$), which you can use to create any permutation. These permutations are equivalent to a $e^{i\frac{\pi}{2}X}$ and $e^{i\frac{\pi}{2}Z}$ rotation, respectively. These are not real matrices, but do indicate much more structure.