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