Would "layered" radio interferometry work? tl;dr - Is splitting up the process of interferometry as shown in the diagram possible, and if so, is it more efficient and/or easier than traditional methods?

I have been doing some research into radio interferometry, and I had a question about it - I know that we can combine the signals from multiple telescopes into one "image" using interferometry, but what if I was able to do that multiple times?

Let me describe this a bit better. Say I have nine dishes. I arrange three of them in an equilateral triangle, and then I do the same with the other 6, and then, I arrange those three groups of three into a larger equilateral triangle. Then, I combine the signals from each three of the telescopes, and then I combine the three resulting signals. Here's a diagram that might help - the blue dishes are the "telescopes", and the red and green boxes are the "processors" or where the signals would be interfered.



In principle, would this work? And, in the context of radio astronomy, would it be easier to combine or interfere only three signals rather than nine?
Absolutely this can work as can be seen in  The Murchison Widefield Array or MWA in Western Australia. This is essentially exactly how it works. You definitely do loose information (the first stage of combining antennas significantly reduces the field of view, and the number of interferometric baselines you get is reduced).
From this page:
Frequency Range            70-300 MHz
Number of Receptors        2048 dual-polarization dipoles
Number of antenna tiles    128
Number of baselines        8128

There are 32 receptors per tile (2 polarizations for $4x4=16$ elements), and $N = 128$ tiles. The number of baselines is $N(N-1)/2 = 8128$.
The advantages is a reduction in the data rates, so reduced digital processing. Note the first stage would be treated as beam forming - you cannot do interferometry between the first stage and somehow do interferometry between the results. Ie the first stage is summing the voltages (with appropriate phase corrections) and the second stage would be interferometry.

One of the tiles making up the 32T, a prototype instrument for the Murchison Widefield Array (source)

An MWA antenna consists of a four by four regular grid of dual-polarisation dipole elements arranged on a 4m x 4m steel mesh ground plane. Each antenna (with its 16 dipoles) is known as a &quot;tile&quot;. Signals from each dipole pass through a low noise amplifier (LNA) and are combined in an analogue beamformer to produce tile beams on the sky. Beamformers sit next to the tiles in the field. The radio frequency (RF) signals from the tile-beams are transmitted to a receiver, each receiver being able to process the signals from a group of eight tiles. Receivers therefore sit in the field, close to groups of eight tiles; cables between receivers and beamformers carry data, power and control signals. Power for the receivers is provided from a central generator.