  Using a new approximate strong-randomness renormalization group (RG), we
study the many-body localized (MBL) phase and phase transition in
one-dimensional quantum systems with short-range interactions and quenched
disorder. Our RG is built on those of Zhang $\textit{et al.}$ [1] and
Goremykina $\textit{et al.}$ [2], which are based on thermal and insulating
blocks. Our main addition is to characterize each insulating block with two
lengths: a physical length, and an internal decay length $\zeta$ for its
effective interactions. In this approach, the MBL phase is governed by a RG
fixed line that is parametrized by a global decay length $\tilde{\zeta}$, and
the rare large thermal inclusions within the MBL phase have a fractal geometry.
As the phase transition is approached from within the MBL phase,
$\tilde{\zeta}$ approaches the finite critical value corresponding to the
avalanche instability, and the fractal dimension of large thermal inclusions
approaches zero. Our analysis is consistent with a Kosterlitz-Thouless-like RG
flow, with no intermediate critical MBL phase.
