Let $g_1,g_2$ be Riemannian metrics on a non-compact manifold $M$ such that $d_1,d_2$ are their distance functions respectively. They are quasi-isometric if there exist constants $A\geq 1,B\geq 0$ such that $$\frac{1}{A}d_1(x,y)-B<d_2(x,y)<Ad_1(x,y)+B$$ for all $x,y\in M$.
If $B=0$, they are bi-Lipschitz. What I want to ask is that when $g_1,g_2$ are quasi-isometric but not bi-Lipschitz, can I control the set $F_C$, which is the set of points where local bi-Lipschitz fails with respect to $C>A$, defined as $$F_C:=\{x\in M|g_1(u,v)>Cg_2(u,v) \text{ or } g_1(u,v)<\frac{1}{C}g_2(u,v)\text{ for some }u,v\in T_xM\}$$ For sufficiently large $C$? If each connected component of $F_C$ is contained in some compact set, it would be great.
Motive of this question is that when two metrics are bi-Lipschitz, we can make them quasi-isometric but not bi-Lipschitz by changing one of two metrics. To do so, we can 'make balloons' like the following figure:
which is altering the metric within compact sets while the resulting set is uniformly bounded.
I want the converse of this, but weaker statement or some properties of $F_C$ would also be very helpful.