You put the cold side of the phase change on the internal cooling loop, step up the external cooling loop as high temp as you can and then circulate that through the radiators. You might even do this step up more than once.
Imagine the data center like a box, you want it to be cold inside, and there’s a compressor, you use to transfer heat from inside to outside, the outside gets hot, inside cold. You then put a radiator on the back of the box and radiate the heat to the darkness of space.
This is all very dependent on the biggest and cheapest rockets in the world but it’s a tradeoff of convenience and serviceability for unlimited free energy.