Why do I get a "transaction already started" error when calling a procedure?

This often happens if the procedure mixes INSERT or UPDATE and SELECT statements, and AUTOCOMMIT is turned off. This is quite a common question from people using Perl where the script connects with AUTOCOMMIT turned off. For example:

my $dbh = DBI->connect($ENV{DBI_DSN},
                        $ENV{DBI_USER},
                        $ENV{DBI_PASS},
                        {
                            RaiseError => 1,
                            AutoCommit => 0
                            }
                        ) || die "Database connection failed: $DBI::errstr"