Definability of mixed period maps

We equip integral graded-polarized mixed period spaces with a natural $\mathbb{R}{\mathrm{alg}}$-definable analytic structure, and prove that any map associated to an admissible variation of Hodge structures is definable in $\mathbb{R}{\mathrm{an,exp}}$ respect this structure. As consequence we re-prove the zero loci normal functions are algebraic.