Are there refrigerators with a remote compressor?

Commercial refrigeration units are available which place the compressor in a remote location. There's a good summary of the pros and cons of such a setup at Katom Restaurant Supply which is rather comprehensive.

One thing to consider is that a typical home air conditioner is about the equivalent of a refrigerator with a remote compressor.

A post two years ago on Houzz website has about 20 comments with nothing conclusive with respect to residential systems, but the expected reference of commercial/restaurant sources.

I'd expect that you'd have to pay the premium for a commercial unit.

I have never seen a residential unit with a remote compressor/ condensing unit.

You could convert a standard refrig into a split unit similar to a regular household AC unit. You'd have to remove the refrigerant, disconnect the compressor, fan and condensing coils from underneath the refrig and install them in the basement. Then reconnect the tubing and electrical and reinstall the refrigerant (would need a pro to remove and reinstall).

Obviously, this would be a real challenge for a DIY but it could be done if you really wanted a conversation piece.

Once upon a time (early 1900's) refrigerators were built-in cabinetry with insulation and refrigeration, and the compressor was often located in the basement just as you'd like.

I don't think there is an regular household appliance-type refrigerator built this way, but there are commercial refrigerators made with the compressor separate, and built-ins are still made for the commercial and high-end residential market.

If you were going to modify a regular household refrigerator, it would be a bit of a project and you'd probably have trouble getting hold of the refrigerant without a license. It would also be a challenge fixing the refrigerator in place so you don't tear off the refrigeration lines when you move it.