my @path = '.';
while (<DATA>) {
    chomp;
    my ($prefix, $node) = /^((?:\| )*[|`]-- )(.+)/
        or next;
    $#path = length($prefix)/2 - 1;
    $path[-1] = $node;
    print $prefix, join("/" => @path), "\n";
}
[download]

my @path = '.';
while (<DATA>) {
    chomp;
    my ($prefix, $node) = /^((?:\| )*[|`]-- )(.+)/
        or next;
    splice @path, length($prefix)/2-1, 0+@path, $node;
    print $prefix, join("/" => @path), "\n";
}
[download]

It avoids the addition of magic to @path, and it combines two operations. The latter usually makes something more readable, but it's unfortunately not the case here.

Speaking of readability, that regex is rather hard to read. I had something like that initially, but I switched to split because it was much more straightforward.

my @path = '.';
while (<DATA>) {
    chomp;
    my ($prefix, $node) = split(/(?<=-- )/, $_, 2)
        or next;
    $#path = length($prefix)/2 - 1;
    $path[-1] = $node;
    print $prefix, join("/" => @path), "\n";
}
[download]