An affine Hadamard family $H(\mathcal{R})$ stemming from a dephased $N\times N$ complex Hadamard matrix $H$, associated with a subspace $\mathcal{R}$ of the space of all real $N\times N$ matrices with zeros in the first row and column, is the set of matrices $$\Big\{H\circ{\rm EXP}\left(iR\right):R\in \mathcal{R}\Big\}.$$