ORIGINAL: Prince of Eckmühl
It truly does appear that NO examples of such vehicles armed with the KwK 36 were shipped to the Afrika Korps. The best fix might just be to put a IIIf-DAK version in the estabs that has the KwK 38 instead. Of course, you could always include the IIIg as a part of a light panzer company. Please, please, please!
PoE (aka ivanmoe)
I guess the term "KwK 36" in the estabs refers to the 37mm gun, which was designated "3,7cm KwK 35/36 L/46.5" or "KwK L/46.5", probably just dubbed "3,7cm KwK".
From August 1940 until 1942 all 3,7 cm Ausf. E, F and G had been rearmed with the "5cm KwK L/42" gun, or - if you will, with the KwK "38" L/42 - the short barreled 50mm gun.
Officially, and at least initially, all PzIIIs (prior to version H) sent to North Africa were designated Ausführung "G (TP)", TP=Tropen ("tropical"), as they were supposed to be equipped with air/dust filters designed for the desert environment.
The equipment did not reach the first tank contingent in time though, so they had to start their march to the East without those filters, causing a significant number of breakdowns even before having met the enemy.
PzIIIs with the long barreled gun (5cm KwK L/60) were dubbed "Mark III Special" by the Brits. In early engagements in North Africa, tanks with the short 50mm L/42 gun performed okay against most British tanks, they just couldn't crack Matildas in a frontal attack, and not above a range of 300 meters, so they used to flank them and get close. In general, the long barreled version provided for more accuracy and enhanced range.
According to what I've read so far, it's hard to verify what versions had been shipped to North Africa, as after continious upgrading / rearming (to meet the G, H or - later on - J standard), the vehicle pool was rather a wild mix that had been upped to at least G and later on H-level (both with the short 50mm gun). Panzer IIIs with the 37mm gun were considered to be underarmed, even for the African theater. In 1941, the vast majority over there was equipped with the 50mm short version, with the rest being equipped with the 50mm long version.
Very few 37mm Pz III may have served as command tanks, or in rear guard duty. Special command tanks (Panzerbefehlswagen) based on Ausf. D, E, H (with fake guns some with working guns) had been produced, so I don't see why they should have used 37mm PzIIIs there, especially since they didn't have the more powerful radio equipment that came along with the command tanks.
Also, the Germans were really good at retrieving damaged/knocked out tanks in North Africa though, so it didn't make sense to employ underarmed 37mm PzIIIs, as they could only fight Crusader tanks (AFAIK). At around the battle of Sollum, the Germans recovered 7 of 9 abandoned/knocked out Matildas and incorporated them into their ranks. The Germans retrieved around 7 out of 10 knocked out German tanks for the most part of the campaign, IIRC.
Parts of the Panzer III Ausf. J production were sent to Russia and North Africa in late 1941/early 1942, to replace the losses on these theaters (appr. 1,400 Pz III in 1941, according to Jentz.... with the losses in Africa being a minor factor, compared to the massive losses in Russia).