This issue has nothing against HTTP method used. Some time ago browsers (first one was Safari) started to block 3rd party cookies and you can do nothing with it - you had to force users to change privacy settings, what is impossible.
Note, that browsers allow you to set 3rd party cookie if you have visited the site as 1st party and created a cookie. I have seen solutions with a kind of ping-pong, like user with no network_id cake is redirected to landing page, that sets network_id cookie and pings user back to referrer page. This works but is ugly. I am not sure if it is 100% legal. Note, that iframes do not work here - this have to be browser _top window...
Some time ago there were possible solutions with using Etag meta field, but this does not work (at least with Safari) no more. Moreover, using this or similar practises could make you pay lots of $$ for user privacy violation.
As far as I know, there is no easy, clean and legal solutions to track users between sites.