I wouldn’t count France or India in WWII as triple fronts: there were French and Indians groups fighting on both sides but no groups fighting against both sides.
China, on the other hand, would probably qualify. The Nationalists and the Communists were fighting each other and both were fighting against the Japanese.
The Battle of Mers-El-Kebir was fought between Britain and Non-Vichy France, and later on, the Germans invaded Vichy France- so the Vichy French were fighting The Germans (again) and the British and the Free French- both of whom were fighting the Germans and the Vichy French.
^
You forgot the Syria-Lebanaon campaign, carried out by the British 9th Army.
I knew someone who fought in N Africa, Italy, Burma and there, said none of those three fronts had the viociousness of that campaign.
The history of Finland during WWII is interesting. They first fought against the Soviet Union with some military aid from Great Britain and France; then with support from Nazi Germany against the Soviets; and finally after an armistace with the Soviets they fought against the Nazis. So during the course of the same war they fought both as allies and as enemies against two powers that were at war with each other, although not quite simultaneously. I wouldn’t exactly call this “switching sides,” since Finland’s only interest was maintaining its own independence.
But the French fleet at Mers-El-Kabir was under the control of Admiral Darlan, who was a Vichy supporter. At least, that’s how I read the wiki article.
The Vichy regime wasn’t established until a week or so after Mers-el-Kebir; so whilst Darlan might have had Vichy leanings, the fact was the UK did engage in an act of war against an allied nation (France) by sinking their Mediterranean Fleet (to stop the Germans getting their hands on it), as France hadn’t yet split up.
So, you had Frenchmen fighting the Germans (The Resistance movements, plus various French Colonies which refused to acknowledge the surrender), Frenchmen fighting the English (Mers-el-Kebir), and the English and the Germans fighting each other.
Initially I thought this might be said this of the Spanish/Tlaxcalans/Aztecs, until I realised that at the time Tlaxcalans and Spaniards were fighting each other the Aztecs weren’t hostile to them.
However, given the tribal nature of Native American society and the belligerent nature of European powers I’m fairly sure an example can be found in the Americas.
The Sengoku period of Japan was also one giant mess, it wouldn’t surprise me to find three-way mutual hostility there.
Dutch army and navy did fight the Japanese – though quickly overrun. The admiral commanding the ABDA fkeet was a Dutchman – until he was fish food.
The Dutch East Indies Air Force relocated to Australia and continued to provide token offensive attacks on the Japanese throughout the war as well, FWIW.
During the Napoleonic Wars there was so much switching of sides, I would guess some three way fighting must’ve went on
This is what prompted my original question - I read that as saying that the British fought the Free French. The fact that the UK and Free France may have had competing goals and tactics doesn’t mean that they were fighting each other. They were in a perhaps uneasy alliance, but that’s not the same as saying they fought each other.
And with respect to Mers-El-Kabir, if it preceded the creation of Vichy France and De Gaulle’s split from Vichy, I don’t see how it can be interpreted as a British attack on Free France?
Well, given that the French Fleet at Mers-el-Kebir hadn’t officially gone over to the Germans- or actually done anything at all, in fact- I think a case can be made for the British attacking France- a France that was still fighting the Germans, despite the Armistice (Several French Colonies, particularly in West Africa, decided to keep fighting anyway). As a result of Mers-El-Kebir, the Free French had quite a difficult time getting recruits for a while which makes for an interesting “What-If?” for armchair military historians.
So yeah, it’s a bit of a stretch, but there was at least one situation in which The Republic of France was fighting both the British and the Germans at the same time. Then you’ve got Case Anton, resulting in the Vichy French fighting the Germans and the Free French and the British Commonwealth for another brief period.
If anything, even a cursory study of the fighting between the Free French and the Vichy French should put an end to the “Cheese Eating Surrender Monkey” myth, but alas, no, it’s going to be at least another generation or two before people let them live that one down.
I wouldn’t be at all surprised if France one day decided to invade Liberia or somewhere like that just to remind people that they’re not to be fucked with.
Since several people have forgotten the OP and given examples that are not 3 distinct countries, doesn’t the current conflict in Iraq qualify under these terms?
Sunni insurgents vs shi’ite insurgents vs the coalition
During the japanese invasion of China, there were several timjes when the Communists and nationalist forces were at fighting (at least low level) while officially united against the japanese.
That’s already been mentioned at least twice already. Once more and it’s going to turn into another Meme, you know… 
Texas might be a good example. The Commanche halted the Spanish at Central Texas. Hoping to solidify their claim to the land, the Spanish provided land to Anglo settlers, who fought the Commanche but ultimately didn’t see themselves as subjects of Spain or of Mexico. After securing Texas independence, the Anglos drove the Commanche (along with almost all other Native Americans) from the state.
What about poor old King Harold in 1066? . He successfully fought off the Vikings at the Battle of Stamford Bridge, only to find that England was also being invaded by the Normans. So he then had to march his troops down to the south coast where they were defeated at the Battle of Hastings.
That would only apply, I think, if the Norwegians were also at war at the same time with the Normans, which they weren’t.
The obvious conflict that comes to mind is the 3rd Crusade, or the Kings’ Crusade. Several European kings (who weren’t exactly friends) joined together as a massive army to drive the muslims out of the holy land. The problem is, the kings argued too much and their armies spent more time fighting each other than the muslims, giving the muslims an easy victory.