This is the statement that always comes back to bite biologists. If it's "theoretically possible", biology is probably already doing it somewhere.
When I started grad school (mol bio/genetics), there was a laundry list of things that "never happened in biology". By the time I finished grad school a lot of those items were removed from the laundry list.
And, as I'm sure you're aware, the inability to find something is not evidence that it doesn't exist.
You might imagine that the cell would do this by splitting the two DNA strands and sending some molecular machine down each one to replicate it. That is what it does - kind of. See, the tricky part is that the two DNA strands are pointing in opposite directions. The heads and tails of the nucleotides (A, C, T, or G) in each strand are pointed in opposite directions.
You might think that, if evolution can create a machine that works in one direction, it could create a machine that works in the other direction. Then, one could be used on each strand in parallel. Back in the day (~40 years ago), this is basically what everyone assumed must be happening.
But that's not what happens. We only have a machine that goes in one direction. People spent many, many years looking for these little molecular machines, but only ones that went in the same direction were found. None that went backwards.
So for the backwards strand, it's duplicated in a really convoluted process. Basically, instead of copying it all in one shot, it has to repeatedly jump ahead, work back, jump ahead, work back, creating a bunch of little DNA fragments. While it does this, all the little fragments have to be tied together. It's a very strange process.