Slitherine

When playing as Rome, what triggers the Marian reforms of the army? In my first playthrough it triggered somewhere around the 130 turn mark. However in the following playthrough it has not triggered and I am past turn 145. I looked in the game files and it appears that all you really need is 3 or more provinces in order for the reforms to trigger. In my second playthrough, I have 4 provinces and have for a while, plus plenty of gold, money and manpower, yet the reforms haven't triggered. I've looked through the manual as well, but couldn't find anything specific.

Is there something specific I need to do in order for the reforms to trigger, or is it just a random chance after you achieve 3 provinces?