Radio telescope arrays record the full waveform of the electric field as a function of time, which is possible with fast electronics. For optical frequencies (≈ 500 THz) this is far beyond what's doable today. You could detect only the phase, relative to some reference, which changes on timescales of the coherence time $tau_c$ of the light. According to the Wiener-Khinchin theorem the coherence time is the inverse of the bandwidth $Delta nu$ of the light. For blackbody radiation from stars $Delta nu$ is still hundreds of THz. You can filter it to arrive at a lower bandwidth, at the cost of losing most of the intensity and therefore resolution. I'm not aware of any oscilloscope faster than 100 GHz, so if you filter the light down to this bandwidth its intensity is reduced by 3 orders of magnitude.

Phase

The heterodyning method you described works well to compare the phase of the incoming light with the phase of a local oscillator. But for interferometry you need to compare the phase of the light relative to the phase at all the other detectors. This is only possible if the all the local oscillators have the same phase.

But how can you synchronize their phase? You could try to build very stable lasers which can be tuned to an exact absolute frequency (the center frequency of the light you filtered out). If you want to measure for 1000 s the lasers need to have the same frequency with an uncertainty well below 1 mHz. Moreover, they must be at this very frequency for the whole measurement duration – usually the frequency stabilities of lasers are given on timescales of a few milliseconds. The best stabilization I know of achieves $10^{-16}$ accuracy over a timescale of seconds. ^[1]

Alternatively you could use one global oscillator as phase reference, i.e. the output of one single laser distributed to all heterodyne setups. Here you basically shift the problem from building a phase-stable interferometer for the starlight to building a phase-stable pathway to distribute the light of the global reference light. In this case you don't suffer so much from intensity losses in the fiber, but still any phase shift will ruin the synchronization. And there are many ways to mess up the phase, like temperature changes or strain.

In both scenarios, if the detectors are on earth they have to look through the atmosphere, which is yet another opportunity to mess up the phase. The refractive index of air depends on its density and composition, which varies spatially and teporally. This phenomenon is called seeing.

Data Sizes

Although I don't think this is a limiting factor, let me say a few words about the amount of data recorded in this kind of measurements. Firstly, there is no point in using a detector with spatial resolution, like a CCD chip. The resolution of the telescope array comes from the distance of the individual detectors, not from the resolution power of an individual detector. Instead one would have a single heterodyne detection setup without any spatial resolution in each telescope. And instead of recording one datapoint with an integration time of 1 minute you need to constantly measure the phase, which is changing very fast, as mentioned in the beginning. The Keysight UXR1104A Infiniium measures 256 GS/s with 10 bit resolution. This data rate can be easily transferred through a single optical fiber. I don't know how to store it, but people at LHC probably solved this problem as well.

What Instead?

One can make use of the intensity fluctuations in thermal light as it was done by Hanbury Brown and Twiss. ^[2] These measurements don't require to measure the phase of the light itself, but on the random interference of light from different parts of the source. Therefore it is not affected by atmospheric turbulences. The intensity varies on timescales of the coherence time $tau_c$, and this is the level of synchronization one needs. 100 GHz synchronization is much easier than 500 THz as required for phase synchronization.