This gives the final result I think you want:
my @oldlist = map {chomp; $_} <DATA>;
my @sorted = map {$_->[2] . " " . $_->[1]}
sort {($a->[2] cmp $b->[2]) || ($b->[1] cmp $a->[1])}
map { [split] } @oldlist;
my @newlist = map {shift(@oldlist) . ' ' . $_} @sorted;
print join "\n", @newlist;
__DATA__
a1 p1 q5
a1 p1 q6
a1 p2 q4
a2 p3 q9
a2 p2 q8
a2 p2 q9
__OUTPUT__
a1 p1 q5 q4 p2
a1 p1 q6 q5 p1
a1 p2 q4 q6 p1
a2 p3 q9 q8 p2
a2 p2 q8 q9 p3
a2 p2 q9 q9 p2
HTH
Update: Forgot to mention my output is different from yours, I think it was a typo in your sample output, looking at your sample input.