The rx/tx separation in over-the-horizon radars (OTHR) serves to increase isolation between the transmitter and receiver. The received signal is tiny, as it has not only bounced off the target but done a double bounce off the ionosphere. Doppler shifts are also very small, so one needs to minimise the clutter at zero Hertz, and a direct signal will appear as zero Hz clutter.

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.