On *nix, I don't need to do <<`EOL`, i.e. <<EOL will do

That's not true. Taking the same code,

open(OUTPUT, " > eol_dos.out") or die $!;
print OUTPUT <<EOL;
dir
EOL
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>ver

Microsoft Windows XP [Version 5.1.2600]

>perl a.pl && type eol_dos.out
dir
[download]

open(OUTPUT, " > eol_dos.out") or die $!;
print OUTPUT <<EOL;
ls -1
EOL
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$ uname
Linux

$ perl a.pl && cat eol_dos.out
ls -1
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The sqlplus is working on *nix but not on Windows

No. Like I said in my previous post, your sh shell command works in sh but not in cmd.

I do not want to fork a separate shell as well so I took it off the bash.

But that's you're doing. You're asking Perl to execute shell command

sqlplus -S "/as sysdba" <<"SQLEND"
set heading off
alter session set nls_date_format = 'DD-MON-YYYY HH24:MI:SS'
;

select 'Connected to the database on ' || sysdate
from dual
;
SQLEND
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so it launches a shell to execute it.

Any suggestion on the sqlplus thing for Windows will be very much appreciated.

Yes, stop invoking the shell. Create a pipe to sqlplus and pipe the data to it instead of asking the shell to do that.

Ideally, that would be

my @cmd = (
    sqlplus => (
        '-S',
        '/as' => 'sysdba',
    )
);

open(my $to_sqlplus, '|-', @cmd)
    or die("Can't launch sqlplus: $!\n");

print $to_sqlplus <<'__EOI__';
...
__EOI__

close($to_sqlplus);
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But no-one has gotten around to implementing multi-arg open |-. So you can use

use IPC::Open2 qw( open2 );

my @cmd = (
    sqlplus => (
        '-S',
        '/as' => 'sysdba',
    )
);

local *TO_SQLPLUS;
my $pid = open2('>&STDOUT', *TO_SQLPLUS, @cmd)
    or die("Can't launch sqlplus: $!\n");

print TO_SQLPLUS <<'__EOI__';
...
foo
bar
...
__EOI__

close(TO_SQLPLUS);
waitpid($pid,0);
[download]

BTW, out of curiosity, do you type in your comments/reply typing <p> and </p> manually as well?

Yes. Well not </p>. It's optional in HTML.