Now let's see here...
It has been awhile since I went to school for engines. Teachy said that oil burns blue, and gasoline burns black. Blow a gasket and coolant will come out as white smoke... Now wait a sec, on a air cooled engine? So what have we gotta work with here to make some white smoke? Gasoline and engine oil and a single cylinder engine. unburned gasoline could produce a cloud of white smoke, but how to get that without running it across the piston and burning the stuff? Maybe if the crank case was loaded with extra gasoline and it didn't want to burn anyway (stale) but getting vaporized in there and chugged thru the vent tube... that could do it.
If the engine is cold enough then I suppose that the diluted oil could make it past the rings and produce a haze, but I would expect it to be a delightful shade of blue. Warm it up enough, and it could clear up rings seating better, or the oil is igniting better in cylinder.
One time I was getting rid of some horrible mixture of diesel and gas in a old chevy, that produced a incredible cloud of smoke. mostly blue, but in certain lighting conditions it could have been white. The 283 did what it could to burn the slop... With that thought in mind it is possible that the fuel is lacking in the more burnable parts of gasoline and the base that is left isn't so easily ignited but can be vaporized with heat and compression, producing white smoke!
Just thinking out loud here...