The real Pauli group is the subgroup of $ O_{2^n}(\mathbb{R}) $ generated by products and tensor products of $ X $ and $ Z $ (this deviates from the usual Pauli group in that only real Paulis are allowed, in other words there is no global phase of $ i $).
The real Clifford group is defined to be the normalizer in $ O_{2^n}(\mathbb{R}) $ of the real Pauli group.
The real Clifford group is finite of size $$ 2^{n^2+n+2}(2^n-1)\prod_{j=1}^{n-1}(4^j-1) $$ see https://arxiv.org/abs/math/0001038
According to the same reference the real Clifford group contains all diagonal gates of the form $$ diag((−1)^{q(v)}+a) $$ where $ q $ is a binary quadratic form and $ a $ is $ 0 $ or $ 1 $. The real Clifford group also contains all permutation matrices that correspond to the action of an element of $ AGL(n,2) $ on the $ 2^n $ basis vectors.
These two types of gates are mentioned in the context of a "convenient generating set" for the real Clifford group. Are all real diagonal gates and all permutation matrices in the in the real Clifford group? (I assume not because Toffoli is a permutation matrix that is not in the Clifford group) Or are the diagonal gates in terms of quadratic forms and the permutation matrices correspond to $ AGL(n,2) $ the only diagonal and permuation gates, respectively, in the real Clifford group?