I assume now you could probably use GPS/atomic clocks to keep them in sync rather than a cable.
You can get an idea of the precision of the timing required by considering how far a radio wave travels in a unit of time corresponding to the uncertainty in your time source and comparing that to the repetition period of the waveform. OTHRs typically have large ranges and use long period waveforms to minimise range ambiguity, so the timing requirements probably aren't too bad.
Summary: A passive OTHR should be doable, especially as a cheapish SDR+computer can digitise and process the entire HF band. Front end sensitivity might be a concern? Low phase noise is critical. Phase noise is where many SDRs fall down?
It's always been in the back of my mind that a distributed passive radar would be a cool project. Something like the existing network of hobbyist ADS-B receivers but pumping out timestamped HF samples which could be used for all manner of purposes. GPS could probably achieve a precision of single digit nanoseconds, which might be small enough to consider the receiver outputs as coherent, allowing synthesis to be done.
If a node's network connection couldn't support continuous transmission it would still work to do burst mode, whereby each node might capture the first second of each minute (for example) then have a minute to transmit that data over the network.
It would be cool to try again with caesium or hydrogen maser ...
So people have used them for locating transmitters, but the accuracy is not amazing.
Still very impressive capability available for free to everyone.
The docs mention that the clock is GPS disciplined, but is it the case that the samples are timestamped in a way that can be traced back to an absolute time. (If not, it would be a matter of programming?)