Likely the pump shaft seal has gone.

This is an easy DIY replacement repair. Go to your local dealer and buy the water pump and oil ring kit.

 

This may well be. Where would you expect oil to come out of if a seal is bad? No oil in coolant.

 

I should add that water pump failures have decreased significantly from the origin K1600's

 

Prince's first post explained it and then thingz went backwards. If the splatter on the exhaust is blue...it's the water pump. If it's oil colored then I'd bet on the shaft seal.

 

Your explaining as if coolant or oil is burning on the exhaust. As I said the weep hole hose is dry. There is oil dripping from the bottom of the water pump, and dripping down on to the exhaust shield below. Enough to look out of place after 400km ride. As far as I can tell the oil originates at the top of the water pump and travels down the seam of the water pump. Could not find any indication of oil around the water pump area. I had read in the forum of a possible source of oil in this area being porous castings or over drilled mounting holes for the water pump. I pulled the two left side bolts for the water pump and found both had oil on their threads. I cleaned the holes and bolts with brake clean and blew out. Used a sealant on the bolts and reinstalled. Tomorrow I will go for a ride and see if any difference. I will let you know.

 

You keep saying oil so I'm sure you are properly indicating it's oil.
As mentioned, the bottom of the water pump has a weep hole that can spit antifreeze. The gen 1 and 2 bikes could do that for a bit and then stop without issue but the gen 3 bikes aren't as bad.
If the substance is actually oil, it's either a bad shaft seal or it's coming from somewhere above the water pump and just seeping down the pump.

 

The gen 2 and 3 have a hose installed on the weep hole which is directed through a hole in the shield. Therefore the drips are behind the shield and fall on the road as you travel. Hence nothing to see. Clever those Germans. 😳

 

So after sealing the two accessible upper left water pump bolts yesterday I went for a ride today. Came back and wiped a tissue under the water pump. i picked up a very small amount of oil. Went for another ride and came back and wiped with tissue and found no oil. Did I find the issue, maybe? This 2022 has only 2000km on it so maybe it just needs more riding? Has 5 months of warranty left but it would be hard for a tech to diagnose. Thinking I will just monitor and at worse I have to pay for a water pump some day...or not.