Three omitted values and non-Blaschke point divisors in half-planes

We construct a real meromorphic function $F$ on $\mathbb{C}$ such that $F^{-1}(\{0,1,\mathrm{\infty }\})\subset \mathbb{R}$, while $F$ is not of bounded type in either half-plane. More strongly, for every $a\in \widehat{\mathbb{C}}\setminus \{0,1,\mathrm{\infty }\}$, the $a$-point divisor in either half-plane fails the Blaschke condition. Thus the construction provides an independent negative answer to a question going back to Nevanlinna’s 1925 work that had remained open for over a century. Postcomposition gives the analogous counterexample for any prescribed triple of distinct values in the Riemann sphere. The core construction and proof were generated during an autonomous run of GPT-5.6 Sol Ultra.