Most likely need a rebuild probably a seal that's let go.
Two options rebuild kits are available as common carb used on many Honda bikes, or you could replace them with something more fancy like a set of sports bike carbs or Racing carbs from Weber!
You could send them off to a carb rebuilder if your not too sure how to do this yourself, others have managed to rebuild them here if they haven't been sat with fuel in that's dried up it's just a simple case of take apart replace anything rubber and put back together again and should work again. Although the auto choke is a bit fussy and can fail and getting replacement sensor for it is a mission I've just had to live without it on mi
Do you know where I could find the rebuild kit?
If I remember correctly, Honda Goldwings use the same carburetor. I too need to buy new floats, this will be my third time replacing them, but this time I'm looking for a new set.if its just a little tube sticking out the carb cover (round thing air cleaner is in) then its most likely a bad or sunk float,the floats in these are plastic and rot out after 30 years i guess or at least mine did...have fun trackin down the float,i did,many out of state calls to different parts shops but i found one eventually i think it was for a keihin motorcycle carb that had the same part numer as keihin had for our preludes...sounds easy enough..lol...sorry dont remember any numbers or websites because it was many years ago im sure ive gotta thread on here somewhere about it...