We start with:

@arr = split /(?=(&&|\|\|))/, $str;
# @arr = ('<chap', '&&', '&&<book', '||', '||<table', '&&', '&&<list')
+;
[download]

Use non-capturing (?:...) instead of the capturing (...):

@arr = split /(?=(?:&&|\|\|))/, $str;
# @arr = ('<chap', '&&<book', '||<table', '&&<list');
[download]

But neither are needed here

@arr = split /(?=&&|\|\|)/, $str;
# @arr = ('<chap', '&&<book', '||<table', '&&<list');
[download]

Are you sure you want to use (?=...)?

@arr = split /&&|\|\|/, $str;
# @arr = ('<chap', '<book', '<table', '<list');
[download]