Shocked I am, shocked!   No one remembered '..' ?

    while( <DATA> ) {
        if( m/#BG/..m/#EN/ ) {
            print "$.: $_";
        }
    }
__DATA__
 blah blah blah
#BG blah blah blah
 blah blah blah
#EN
 blah blah blah
 blah #BG blah #EN blah
 blah blah blah
[download]

will output

2: #BG blah blah blah
3:  blah blah blah
4: #EN
6:  blah #BG blah #EN blah

and as a one-liner Perl

perl -ne "print if m/#BG/..m/#EN/;" anon03.dat