There's no "magical gateway" and time isn't getting unhinged, it's just lazy writing. Brandon had to juggle with different timelines, and didn't put enough pointers in each one to remind the readers of this as the book progressed, the way Jordan would have done in earlier books in the same situation (in COT, for example). There are pointers, just a bit too far apart (some are only in TGS) for the average readers (forum people, who've discussed the timeline a lot in the past, were not confused by this - we all remembered Perrin was at least a month behind)

In TGS shortly after Malden, Perrin saw Rand still in Tear at the manor, before he captured Semirhage.

Toward the end of TGS, Rand at Nynaeve's request located Perrin with the colour swirls. He was on the Jehannah Road, near the statue, with Galad (we now know Rand saw the trial scene).

Cadsuane came or sent for Tam a short while after the trial. A few days later, Perrin saw Rand on DM.

Tam was seen in Tear at the beginning of the book, because Rand's storyline in TOM is set after his epiphany. Loosely speaking, Perrin was already in the Forging scene/battle with Mesaana around this time, and Tam had left him.

The same thing happened with Elayne (but this was more confusing, and many people didn't even notice it...). She was on a different timeline in her scene with Egwene (post reunification) than she was in her scenes with Mat. That's why she never refers to the Seanchan attack or the reunification of the Tower in her scenes with Mat, or why she didn't think of the BA purge when she went to the dungeons (and why the BA were still there, and Egwene had not sent for them and executed them). It had not happened yet, it was weeks before. When Joline and co. left for the Tower, the reunification was still upcoming, and the rumour it had happened was wrong. When they arrived to TV, it had taken place for real.