"Life on the Tether", a novelette by Mark Wheeler in The Magazine of Fantasy & Science Fiction, August 1983, available at the Internet Archive. As far as the ISFDB knows, the story was never reprinted, and the author never published anything else.
The story is narrated by Maura, the female twin. It is definitely science fiction; there is futuristic stuff like autodocs, and the conjoined twins are the unintended result of a scientific experiment:
Dirac and I were the result of an experiment to breed a genius. Genetic researchers designed us—well, one of us, anyway. When the egg was implanted into Helen, the host mother, it divided by some fluke, and then there were two of us. They almost aborted us and started again, being unsure of how the splitting process might affect their carefully laid plans, but the prospect of creating twin geniuses was something Campbell and Pele couldn't turn down. So the two embryos matured side by side in the womb. It wasn't until shortly before they wee due to be born that analysis showed certain irregularities. The effects of these flaws—we've been told they were caused by irregular splitting of the cell—was such that neither child would survive long once born. Further investigation suggested that the flaws were different in each of us, and in fact dovetailed, and that, as a single organism, we might be able to function normally. So they dug us out by Caesarean, and when we were strong enough they performed extensive surgery and linked us via the umbilical. The hairline scars that crisscrossed our bodies were virtually invisible after twenty-two years.
I think the two are living/working in some sort of research lab, or possibly a space station.
Nothing like that. The boy is a musician, the girl a mathematician:
Dirac was a natural musician. And 1 had achieved recognition at fifteen for solving—rather proving—Fermat's Last Theorem, which had been puzzling mathematicians for quite some time. It was annoying to think that the Nobel had gone to Campbell and Pele for making us possible.
There is a crisis in the facility of some sort that I cannot recall. What I do recall is that the brother and sister argue over the best course of action.
The crisis is revealed in the first line of the story:
We're going to have a baby, Dirac and me.
The autodoc came back with the sad news today during our weekly checkup. It whirred and chugged and the probe withdrew from my side, snaking away into its sterile sheath, and the printout came up on the screen. It was enough to take my breath away. Dirac had tears in his eyes. Up till that moment it had been a usual morning, with us laughing and joking over breakfast and on our way up there to the room next to the bathroom where we stored the autodoc, but now a heavy weight had descended, like a thick and veiling cloud over our thoughts. I reached out to touch his cheek reassuringly with a finger, but he gently stopped me and our eyes locked for a few seconds. We learned to read each other's thoughts pretty well over the years, and I can tell you that we hurt.
"We should have been sterilized," Dirac said after an awful silence, one of the longest I've endured since childhood, when we would sit naked watching each other for hours, not touching, but exploring with warm gazes. Now his words rung out like a tenor bell across the canyon, ringing out the absolute truth: we should have been sterilized, long ago. But everyone thought we could be trusted.
She squeezes so hard that the blood flow between them is interrupted and both begin to weaken and fight to remain conscious. It's then we learn that the sister is the physically dominant one, as her brother collapses first and blacks out entirely.
That was in a childhood fight, when they were nine or ten:
"I'm going to kill you," I screamed. On impulse I grabbed at the gray umbilical which was writhing around us in the water and squeezed it with all the might in my young arms. In a few short seconds Dirac stopped struggling, and even through the water that swilled over him I could see the terror in his face. He began to wail at me to let go.
"You'll kill us both, idiot. Oh, let go, Maura, please. I'm sorry," he yelled as hard as he could with me sitting on him.
"I don't care. You'll die before me, and then I'll let go and stay alive and be able to live alone like normal people and have my own friends in a big school. I hate you Dirac, and I hate being attached to you all the time."
Then Dirac couldn't speak, partly because of the water washing over his face as he struggled for air, but mostly because of my grip on the umbilical. I felt a little dizzy myself, but the effect it was having on my brother was quite astonishing. His normally apple red cheeks paled almost instantly and bloated vulgarly, and his eyes took on an odd stare. He seemed to be having difficulty moving.
My anger dissolved into fear, and finally I let go of the cord and got to my feet to stand on the mud and shingle shore waiting, hoping he'd recover quickly. In a few minutes he was able to stand. He began to cry pathetically, with his arms draped at his sides like a rag doll with starched legs. Somehow that made everything all right.