  In Kenig and Toro's two-phase free boundary problem, one studies how the
regularity of the Radon-Nikodym derivative $h= d\omega^-/d\omega^+$ of harmonic
measures on complementary NTA domains controls the geometry of their common
boundary. It is now known that $\log h \in C^{0,\alpha}(\partial \Omega)$
implies that pointwise the boundary has a unique blow-up, which is the zero set
of a homogeneous harmonic polynomial. In this note, we give examples of domains
with $\log h \in C(\partial \Omega)$ whose boundaries have points with
non-unique blow-ups. Philosophically the examples arise from oscillating or
rotating a blow-up limit by an infinite amount, but very slowly.
