The transformers in the power supplies provide 'isolation' between the input and output - which means you can connect the outputs together, even when the inputs are on different phases.

Are you planning to build such a machine for your personal home use? If so you should be aware that (a) you might find server hardware hasn't thoroughly tested compatibility with things like suspend; (b) you might find games haven't thoroughly tested compatibility with multi-GPU setups; and (c) you might find the idle power consumption is 200W or more, even while doing nothing.

It's for personal use, though it would not run any games, it would be for running offline inference and other experiments. Probably not a smart purchase, but a fun one...

That is good to know multiple phases can work. Perhaps there would still be a fire risk in case of a short? Like somehow bridging the circuits > breakers don't trip?

Keep in mind GPUs (and the rest of the computer) run on DC, not AC, so there is no phase by the time it comes to your computer. The PSU will step down the AC to the right voltage and then rectify it into DC, and they do that independently so whatever phase they started with shouldn't matter.

Something to keep in mind though is that (at least with consumer-grade PSUs) it is not safe to simply tie the outputs together, even if both PSUs produce 5V, 12V, 3.3V, etc. The voltages will be slightly different and connecting them together will cause current to flow back into one of the PSUs.

You can still use this setup though, the key is that the GPUs do not (or should not) connect the motherboard voltage provided via the card slot to the voltage provided via the power connector. This detail allows you to safely power the motherboard from one PSU and power the GPU from another one, you just have to be careful not to mix connectors on the same card between different PSUs (if it has multiple). Additionally the motherboard should be entirely powered from a single PSU.