Hazy filament sounds ominous, like fish eggs that havent been fertilised or have died and are growing fungus.
http://www.fishdoc.co.uk/disease/fungus.htm
if they are infertile and growing fungus I would get rid of them. I dont know if fish, koi/goldfish, eggs are that big but would have thought that a fertile egg had a dark core. You might try a google for XXX eggs.
With only 3 koi I suppose you could have all females and that they might just dump eggs but I have seen that they occasionally reabsorb them, I am sure 2ft koi should be capable of breeding.

Are they laid singly or in a clump? What colour is the bit in the centre?
I'm afraid I don't know what they are but the info might help others.

Thanks for info so far. Looks like my detergent post and this post could be related after all.

The eggs are attached to blanket weed singly and fairly evenly spaced out. The centres are white and firm to the touch.

Although the fish show no obvious signs of distress I notice they are not feeding too well. Water temperature is about 15C which is cold for this time of year but not cold enough to stop feeding.

So I have just run some water tests. Ph and ammonia are OK at 7.5 and 0.8 mg/l respectively. But nitrite at 1.5 mg/l and nitrate at 40 mg/l give cause for concern.

I run a Oase BioTec 5 with UV and estimate water at between 700 and 1000 gallons. Appreciate this is small for 4 fish of such a size but they have just grown to that - not planned.

However, whatever has happened has happened suddenly (like overnight a few days ago), not gradually. Whether exactly coincident with the 'eggs' I'm not certain, but it's a fair bet. Prior to this all pond tests revealed a good water quality and the fish looked in excellent condition.

Now I'm in uncharted water, so to speak. Off to do another partial water change (tested tap water - negligible nitrite) and appreciate opinion. Especially the remark by chris11 "spawning" in response to my detergent post

Alex

fish spawn mainly from late may to early june, but can spawn all summer up to late september, we did have a couple of warmish nights about that time, and that is often a trigger for spawning, so its highly likely they have spawned, or at least one has shed its eggs for what ever reason, but the foam is often associated with a spwning and is caused by the males, so keep an eye on the egg cases, you might getlucky, with temps as they are at mo they take about 10 days or so.

Take a look out with the torch at night. See if you can see any spawning behaviour.