Okay, well, there are several major problems with Kritarchy. For one thing, Montesquieu would turn in his grave at the prospect of ALL the branches of government being tied up in someone who is supposed to be impartial. Another is that you really need a well-established set of laws a priori to be refined by the judges on an ad hoc basis, kinda like English Common Law. Hence...
Scenario 1: Australia
If you have a really strict Governor-General during the early penal colony period who believes that criminal behaviour is hereditary, he will probably use his almost absolute powers to punish wrongdoers (maybe by sending them to the interior?). The executive will fuse with the judicial function, and you'd probably eventually get a hierarchy of common-law Governors/Judges throughout the continent. This will probably end up like apartheid, except with two white tiers. Hence, the Free population needs to be much smaller than OTL and paranoid about its position. Judges are all stick and no carrot, so they might be able to promote people to higher castes to keep them mollified - hence corruption and the watering down of the elite. This kinda thing won't survive past the end of the Cold War, if that.
Scenario 2: Sharia Law
Basically the same as OTL Somalia and Iran. PODs include the Army supporting Morsi against Egyptian protesters, followed by dissolution of Parliament or Khomeini going the whole hog and getting rid of the Majlis. But if the Kritarchy emerges before 1920-ish, it'll get overthrown by the British or French and replaced with a nice friendly monarchy, like Jordan or Saudi Arabia.
Scenario 3: French Revolution
Say Robespierre et al don't escalate the Terror (alienating Paris) and keep their power for a few more years. Supposing the general course of the war goes as OTL they could make peace with everybody in the wake of Campo Formio - they'd give up all the colonies, Alsace-Lorraine, Corsica and promise not to export the revolution - then whatever the Parliament is called that week could legislate all the Jacobin reforms and be abolished by the Committee for Public Safety, which would become a kind of Supreme Court with Executive powers, with the Terror becoming a low-scale Nervous Anxiety. Obviously, Sieyes, Bonaparte and maybe Talleyrand would destabilise the regime if pushed, and it remains to be seen whether the state would survive Alt-1848 and Alt-1871, but if there was a propaganda genius and a diplomatic genius in the CPS at all times they could conceivably limp along to today in Switzerland Mode.
Of the above, the least ASB is Scenario 2, because of the populist influence of a religion that prioritises 'submission' above reform. Any thoughts on how to make these remotely workable?