Fields of definition for Belyi pairs

Let the Belyi pair be represented in such a way that the coefficients of the equation of the curve and the Belyi function on it belong to a certain number field. Then this field is called the field of definition of the Belyi pair. A natural candidate for the “smallest” definition field is the true field of moduli, which corresponds to the stabilizer of the Belyi pair in the absolute Galois group. We will discuss the conditions under which the Belyi pair admits the realization over the true field of definition and the obstacles to this.