tommytx - 12:20 am on Sep 14, 2012 (gmt 0)

The only way to do that is for the third party to talk to a page on your site.. and it ain't easy to get them to do that unless you are a big money maker for them. Hell they will do anything you ask if you are sending them a ton of buyers.... otherwise you cut them off.
so one of many methods would be for their "Thankyou" page meaning when they buy on the third party site they will be sent to a thank you page.. On that page is a tiny (even invisible image) on that page that will be called from your site... and you simply record the call for that image as a sale for you on the third party site...
Each sale will record a hit on the image they are pulling from your site... so you will also need a php file to monitor that image to make sure that the same IP does not call the image more than once.. in case they refresh the page or return to the thankyou page again later.
If you need more information on how to set this up.... just say so..
There are many ways to skin a cat... you could even record the IP of the person you sent them then when they buy you again get to see the IP of the person who is accessing your little tiny image.... so you could match the visitor ip to the buyer ip..
Ain't software automation wonderful....