Applying this transformation $$ \begin{align} A_a^{\dagger}A_i-A_aA_i^{\dagger} &=\left(\frac{X_a-iY_i}{2}\right)\left(\frac{X_a+iY_i}{2}\right) \tag{14}\\ &-\left(\frac{X_a+iY_i}{2}\right)\left(\frac{X_a-iY_i}{2}\right) \tag{15}\\ &=\frac{i}{2}\left(X_aY_i-Y_aX_i\right), \tag{16} \end{align} $$
The ansatz becomes $$ \begin{align} \vert\Psi(\theta)\rangle =\exp\left\{\frac{i}{2}\sum_{ia}t_i^a\left(X_aY_i-Y_aX_i\right)\right\}\vert\Phi\rangle. \tag{17} \end{align} $$