Sending a multipart E-Mail using PHP Mailer/wp_mail – not all clients display E-Mail

I have some problems with sending a multipart E-Mail through the WordPress function wp_mail, which is basically just a wrapper for PHP Mailer.

In the OS X Mail.app, the Mail displays fine. In Thunderbird, nothing is displayed, while Google Mail displays an attachment called ‘noname’ with the size of 1 KB.

My guess is that it has something to do with the encoding, since I noticed that the boundary is not displayed in the E-Mail’s header at all, while it appears in the header from other E-Mails I have received. Also, there seems to be some kind of Content-Transfer-Encoding mismatch?

These are the important bits of the header:

To: email@address.com
Subject: Testsubject
X-PHP-Originating-Script: 10029:class-phpmailer.php
Date: Mon, 24 Mar 2014 15:52:59 +0000
From: WordPress <wordpress@example.com>
Message-ID: <ab7dd25d45c2d5be71df1e592bb0ab96@www.example.com>
X-Priority: 3
X-Mailer: PHPMailer 5.2.4 (http://code.google.com/a/apache-extras.org/p/phpmailer/)
MIME-Version: 1.0
Content-Transfer-Encoding: 8bit
Content-Type: multipart/alternative; charset=UTF-8

--1234567890
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 7bit

Text Plain

--1234567890
Content-Type: text/html; charset=UTF-8
Content-Transfer-Encoding: 7bit

<b>HTML E-Mail</b>

--1234567890--

In other HTML E-Mail that get correctly displayed, I can clearly read the boundary from the header. Also, the Content-Transfer-Encoding is different.

Content-Type: multipart/alternative; boundary="--==_mimepart_532c53995fed2_5f733fb4b5966cd0158292"; charset="UTF-8"
Content-Transfer-Encoding: 7bit

In my PHP script, the mail is compiled like that:

$to = 'email@address.com';
$subject = 'Testsubject';

$headers = 'Content-Type: multipart/alternative; charset=UTF-8; boundary="1234567890"' . "rnrn";

$message = '--1234567890' . "rn";
$message .= 'Content-Type: text/plain; charset=UTF-8' . "rn";
$message .= 'Content-Transfer-Encoding: 7bit' . "rnrn";
$message .= 'Text Plain' . "rnrn";

$message .= '--1234567890' . "rn";
$message .= 'Content-Type: text/html; charset=UTF-8' . "rn";
$message .= 'Content-Transfer-Encoding: 7bit' . "rnrn";
$message .= '<b>HTML E-Mail</b>' . "rnrn";
$message .= '--1234567890--';

wp_mail( $to, $subject, $message, $headers );

Do you see anything that strikes you here? Why is the E-Mail body not displayed at all in Thunderbird/Google Mail?

At the moment I am thinking that maybe the server is misconfigured or something like that, but the fact that Mail.app displays the mail correctly does not fit into that.