Georgano's 1968
Encyclopaedia has a picture of what appears to be the same car, with the caption "1912 Austro-Daimler Prince Henry Tourer", the photo attributed to Steyr-Daimler-Puch AG. With that big-bore exhaust and no windscreen (and in this photo no headlights), it looks more like a competition car than a tourer in the normal sense. However, this is clearly not one of the 1910 Prince Henry Trial cars, as there are numerous photos of them around. I have another photo of the rear of this car, but it gives no date.
The same picture also appears, along with the second shown below, in Chapter 6 of Karl Ludvigsen's biography "Ferdinand Porsche - Genesis of Genius", with the caption :
Prince Elias of Parma, here a passenger, ordered this magnificent coachwork from Jacob Lohner for his series-built shaft-drive 22/86 Austro-Daimler. With its cycle fenders, tapered tail and huge exhaust pipes it was one of the most stunning sporting cars of its era.
Frustratingly, again no date is given. However, if it is shaft, rather than chain, driven, it is not one of the earliest production chassis, and maybe the 1912 date is plausible.
Not really much help, I'm afraid.