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Thanks chaps, but after repeated failings on my part, I saw sense, and sent her to Oz at Zed Parts to help rid the gremlin.
Honestly no matter what I did she was not having any of it.

Pete, my real issue was not being able to correctly balance / sync all 3 carbs at the same time - ive done it many many times, but not on the Z13. Narrowed it down to the centre carb: Engine Compression at 175psi - good coils - strong spark - timing spot on - carbs appear spot on - etc etc. Buit almost like I could not get a vacuum, but I have no leaks anywhere, and everything else looks OK.

Spoke to Oz the Z13 Meister, and he says he's never seen anything like it before and also is having challenges setting my bike up: so I dont feel such a numpty.

Would really like to get my bike back on the road before the crappy weather returns.

Cheers Tim

Keep us posted with progress and never, never, never give up!

Sometimes if the carbs have been totally disassembled they do need re-aligning to ensure they are level and square to each other. Kawasaki did make a tool for this but I dare say most have long gone. If they are out of line the operating posts which contain the balancing adjustment will go out of range and a vacuum will not be seen to equal the other two carbs. Bench primary set up with a piece of welding wire or similar can sometimes help with initial balance but making sure the carb bodies are correctly aligned really helps.

Quick update on my sick Z13: even though the issue manifested itself after refitting my rebuilt carbs, oddly the gremlin turns out not to be a carb related issue: Ive run my carbs on another Z13 and all is OK, put a different set of carbs on my bike and get same fault: inability to sync / balance the carbs and sick running isolated to Cyl 3 & 4.

Great compression, timing spot on, carbs set correctly, but same sick running issue with No.3 & 4 - so next set of test will be coils and then igniter.

Probably get her runnning just in time for winter

Cheers Tim

Too bad you had to go to the extent of changing out the carb set to determine the issue but at least you're making headway.
Sounds like you have an issue with ignition, be it a bad coil, or ignitor. I would focus on the coil first. So many bad reports of coil issues. I really don't like where Kawasaki placed the coil up above the engine in the hot zone and little to no air flow up under the gas tank to help keep the coils cool. They generate their own heat due to the inductance across the primary and secondary windings (think of an induction stove element) and then add the heat of the engine.
First check for me would be to take new spark plug and open up the gap to 3/8" and place on the plug wire. Place the side of the plug on the engine to ground it out and fire up the bike. You should have a nice blue spark jumping that 3/8" gap. If not, check the coil, high tension wire, spark plug cap and the connections on the primary side of the coil. A weak primary will show up as a weak spark as well. Too bad you don't have access to an oscilloscope.

Good luck chasing the issue.