On Friday, four ministers were trotted out to tell the media the government is prepared to send up to 750 Canadians soldiers and police officers somewhere in the world to support UN efforts (although none of the four would say where).

But more than 800 Canadian troops are already involved in a real fighting war in Northern Iraq against the Islamic militants known as Daesh. And that war has just become desperately more complicated.

The latest complications occurred last week, not in Iraq proper but in neighbouring Syria where Daesh, also known as ISIS or ISIL, is based. Put simply, it involved two important Western allies in the war against Daesh fighting one another.

One of those allies is NATO member Turkey, which is backed by the U.S. The other is a Kurdish militia known as the Peoples Protection Units or YPG. It is also backed by the U.S.

The U.S. views the YPG as its most effective regional military ally in the war against Daesh.

The Turks, however, treat the YPG as a terrorist group. That’s because of its close links to another Kurdish militia, the Kurdistan Workers’ Party or PKK, which is indeed a terrorist group. Or at least that’s what Canada, the U.S. and other NATO countries say.

Last week, the Turkish army made one of its rare forays into Syria. The stated motive was to drive Daesh from an important border town. The real motive, it seems, was to prevent the YPG from driving Daesh out of that important border town.

To that end, the Turkish forces turned their guns on the YPG.

The Turkish government frets that if the Kurds are able to set up a functioning statelet on its border, they will use this as a springboard to carve out a new country, encompassing parts of what are now Iraq, Iran, Syria — and Turkey.

Given that Kurdish nationalists have been demanding their own country since at least 1919, this is not an entirely unreasonable fear.

Canada enters the equation in that it supports and trains the main Kurdish militia, or peshmerga, in Northern Iraq.

About 830 Canadian soldiers are involved in the Iraq mission, including roughly 200 special-forces personnel acting as front-line trainers and advisers.

The Kurds have their own factions. The regional government of Iraqi Kurdistan is dominated by a party that has worked out a rapprochement with Turkey and that, at different times in the past, has fought against the PKK.

But the alliances and rivalries among these factions are subtle and fluid. In the end, all are angling for the same end — an independent Kurdish state.

This was brought home starkly when Ottawa was told the Kurdish militia it has been so diligently training would not be allowed to play a major role in the final offensive against Daesh forces holding the Iraqi city of Mosul.

The reason, as Canadian Press reported last week, is that Iraq’s central government wants to ensure that Mosul, an important oil city, doesn’t fall under the control of the independence-minded Kurds.

All of which is to say that the Levant, as this part of the Middle East used to be called, is a tricky place in which to operate. There are wheels within wheels, rivalries within rivalries.

While it is tempting to describe the fight against Daesh in this region in black and white terms, this is not the reality on the ground. Daesh may be irredeemably awful. But under examination, the white hats fighting it turn out to be coloured different shades of grey.

So good luck with what the government is calling its peace ops. Getting back into the world of UN peacekeeping is not a terrible idea — although, as Defence Minister Harjit Sajjan has noted, there is often little peace to keep.

But let us not forget that Canada is already tied up in a dangerous shooting war, one that is getting trickier by the day.