I find soft eggs like this maybe 3 or 4 times a year but this one is hard. There is 2 female ducks. One is 7 and the other is 6 years old.
They eat organic layer feed and have oyster shell available. When I find a soft egg I add 1/2 teaspoon calcium supplement to the feed. They get nutritional yeast in the feed anout once a week. They run around my entire yard eating bugs worms slugs and weeds.
You could be seeing symptoms of a duck with an infection, you might need to try an antibiotic for a week and see if that helps.
Feed stores sell antibiotics such as tetracycline, a broad spectrum antibiotic for about $10
The other cause could be a lack of nutrients such as;
calcium, zinc, fiber, vitamin B1, magnesium and phosphorous.
The easy way to make sure is with a multi mineral/vitamin from your feed store or even human pills dissolved in the waterer will work.
I feed my ducks "grower" since I have males in the flock, and then supplement with oyster shell on the side. This worked great for years. The ducks would come out of the duck house in the morning, and you could almost always tell who was laying by who ran over to the oyster shell and started eating. But, then I got some new ducks. For some reason, these ducks did not eat oyster shell the same way, and I was getting a lot of soft-shelled/shell-less eggs. So, I put oyster shell in their house, as well as in their yard, and *POOF!* no more soft-shelled eggs. Those ducks just needed access to the oyster shell all night.
I do sometimes get some calcium deposits on the outside of my duck eggs, but I don't think that's a problem--I might be wrong, though!
Looks to me like the hen started to form a shell and an egg collapsed then that shell got stuck to the next egg or she may just wanted to give you a handle to pick the egg up with. I have 6 year old chickens that skip a day or three laying and then will have extra shell bits stuck to the outside of the next egg. So I think yours is an extreme example but not unprecedented.