The internally consistent argument is that these items are, like many other luxury goods, designed to signal status (or, more rarely, to elicit a response of much more fundamental appreciation, the sort a fellow watchmaker might have for a Patek Philippe). Diamonds are not unique in this regard, because many people prefer natural/real over artificial/synthetic/fake when it comes to art, antiques, luxury items, etc.

If you approach this from the perspective that you’re spending more money for imperceptible characteristics encouraged by a massive conglomerate, then sure, it’s obviously irrational. But if you look at it from the perspective of signaling or status seeking, it makes sense to pursue the “real” item over the “fake” one.

If you buy a synthetic diamond, the only people you’re signaling favorably to are those in your particular ingroup with respect to opinions on De Beers. If you buy a real diamond, you are signaling to a much wider set of people. It’s not fiscally rational, but it’s not supposed to be.