It sounds like a two-bit operation if things were that messed up. I wonder, from the sound of it, if some of this was done "online" and involved mailing frozen embryos if it was a "worldwide operation". If so, that might account for how there could be such mix ups as opposed to going to a local clinic. Packages arrive and aren't sorted or stored correctly, essentially. Just a guess.
Our twins were also born via IVF, but all of our embryos were frozen (as my ovaries hyperstimulated from that one cycle we did, so we had to wait) - that might be more where the "test tube" term comes from. They place three embryos in each straw, or pipette, and they are then stored in liquid nitrogen. And yes, a LOT of checking is done over and over to make sure the right embryos are stored and placed into the right mother when the time comes. (And there are a lot of legalities.) With ours, as I was on the table, after the doctor placed the embryos into my uterus, they then took the pipette and examined it under a microscope to make sure there was nothing left inside, which is part of the standard procedure.
We used a top clinic, deliberately. It wasn't always convenient as it was a distance from my home, but that's where I wanted to go, and in retrospect, it was the right decision. As I became pregnant on the first try, we had other embryos in storage for many years. (I had a complication with pregnancy that made another pregnancy unsafe.) Rather than discard them, we later gave them to another couple. They flew, frozen, across the country. That couple used all of them and successfully gave birth to other children, the point being that 100% of our embryos survived the thaw, and of those children that were born, all were 100% healthy. That is a testament to the care that was taken by the clinic we used. Not all embryos are of freezable quality, so they have determinations to make during these processes. The reason I am saying this is for those reading who may be thinking about doing IVF: go to the best clinic you can.
I am very sad for this couple, and somewhat enraged that this clinic was so careless with these potential lives. Clinics are regulated. I suppose we'll be hearing details of what happened in the days to come. If it's awful, it's going to give all of them a bad name, when that's not really the case.
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I'm happy for you that you were able to have your children. How wonderful to offer your embryos to other couples - I didn't realize this was done, truly a beautiful thing.