If you receive a “synchronization error” it would likely mean that your smtp client/script is sending the data before exim has sent it's welcome message. This is used to prevent script spammers from spamming you, as their scripts dont' usually wait for the welcome message.
Info: What does SMTP protocol violation: synchronization error (next input sent too soon) mean? SMTP is a ``lock-step'' protocol, which means that, at certain points in the protocol, the client must wait for the server to respond before sending more data. Exim checks for correct behaviour, and issues this error if the client sends data too soon. This protects against malefactious clients who send a bunch of SMTP commands (usually to transmit spam) without waiting for any replies.
Possible workaround for your exim.conf, top section:
smtp_enforce_sync = false