Yuk!
Just been having another look under the sink with a view to implementing the possible solutions, and started noticing things I hadn't before, mainly because the bit beyond the LHS is inaccessible behind a built-in corner cupboard between the sink unit and the soil stack.
The most important horror is that, courtesy of the (lack of) workmanship by whoever installed the current sink plumbing, there is no fall between the sink and the main soil stack. In fact, if you look carefully, you can see this in both pics, where the weight of water in the trap is pulling the whole thing down in the middle, so the fall is actually slightly reversed!
This in itself is bad enough, though by itself wouldn't be so bad for most modern convenience food cooking, especially as I'm also vegetarian, so there is less solid grease and and more liquid oil going down the sink, but I also happen to prefer traditional tea from a pot, and have always been in the habit of flushing the leaves down the sink. (I've been doing this for 50+ yrs with no real problem up til now, as long as sufficient water is used - but partly in the light of the drought I've been drinking less tea and more coffee which uses less water, and has the added bonus that the grounds are contained by the filter papers and so can be thrown away instead.)
So I'm now wondering if part of the problem might be that tea leaves and other sink gunge is all washing back into the running trap for the standpipes and partly blocking it. I would have thought that the flow from the washing machine would have tended to keep it clear though.
Thirdly, I'm not sure I was right about the Persil Auto, it's the auto bit I'm unsure of, and as I put the stuff in a plastic container and recycle the box, I can't check back. I'll have to check when I'm next in the supermarket. I think it's this one, though I remember it being as being a similar blue packet but with some yellow around the top of it.
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potentially I have a situation of reduced flow leading to more gurgling and churning about of a non-auto powder in the washing machine standpipe, which might explain the excessive froth production in the system, which, because there is no individual trap for it, can then back up out of the dishwasher standpipe.
As a start, I suppose I could try seeing if blocks of wood under the pipes and adjusting the mountings to produce some fall would make any difference, but it's beginning to look like a second trap is required ...