Totally real Thue inequalities over imaginary quadratic fields

Let $F(x,y)$ be an irreducible binary form of degree $\ge 3$ with integer coefficients and with real roots. Let $M$ be an imaginary quadratic field, with ring of integers $Z_M$. Let $K>0$. We describe an efficient method how to reduce the resolution of the relative Thue inequalities $$\mathrm{\mid }F(x,y)\mathrm{\mid }\le K(x,y\in Z_M)$$ to the resolution of absolute Thue inequalities of type $$\mathrm{\mid }F(x,y)\mathrm{\mid }\le k(x,y\in Z)\mathrm{.}$$ We illustrate our method with an explicit example.