  We introduce a class of mappings called vertical quasi-isometries and show
that branched quasisymmetries $X\to Y$ of Guo and Williams between compact,
bounded turning metric doubling spaces admit natural vertically quasi-isometric
extensions $\widehat X\to \widehat Y$ between hyperbolic fillings $\widehat X$
and $\widehat Y$ of $X$ and $Y$, respectively. We also give a converse for this
result by showing that a finite multiplicity vertical quasi-isometry $\widehat
X \to \widehat Y$ between hyperbolic fillings induces a branched quasisymmetry
$X \to Y$.
