  Converting a compressed format of a string into another compressed format
without an explicit decompression is one of the central research topics in
string processing. We discuss the problem of converting the run-length
Burrows-Wheeler Transform (RLBWT) of a string to Lempel-Ziv 77 (LZ77) phrases
of the reversed string. The first results with Policriti and Prezza's
conversion algorithm [Algorithmica 2018] were $O(n \log r)$ time and $O(r)$
working space for length of the string $n$, number of runs $r$ in the RLBWT,
and number of LZ77 phrases $z$. Recent results with Kempa's conversion
algorithm [SODA 2019] are $O(n / \log n + r \log^{9} n + z \log^{9} n)$ time
and $O(n / \log_{\sigma} n + r \log^{8} n)$ working space for the alphabet size
$\sigma$ of the RLBWT. In this paper, we present a new conversion algorithm by
improving Policriti and Prezza's conversion algorithm where dynamic data
structures for general purpose are used. We argue that these dynamic data
structures can be replaced and present new data structures for faster
conversion. The time and working space of our conversion algorithm with new
data structures are $O(n \min \{ \log \log n, \sqrt{\frac{\log r}{\log\log r}}
\})$ and $O(r)$, respectively.
