What piece of technology is able to record the exact phase of microwaves? I read that in VLBI, the signals of microwaves are recorded and then later combined. A famous example of this is the Event Horizon Telescope which imaged the black hole. Presumably, the telescopes recorded the phase of the microwaves. 
This tells me that it is possible to record the specific phase of a microwave, but I haven't seen any technology that can do this. Most oscilloscopes, for example, don't have high enough frequency to resolve such short time scales required to measure microwave phases.
My two questions are simple. Answers to either one would be helpful.
Is there any piece of technology that can measure the phase of a microwave? What technology is used anyways for VLBI to record phases?
 A: tl;dr -- baseband it with beats.
The Problem
If you want to accurately record a high frequency time series, you need to sample the data at an even higher frequency, at least twice the target frequency to be exact.
This is called the Nyquist frequency, and it is a general result of signal processing based on Fourier analysis.
So you want to record a 1 mm microwave signal.
Since all electro-magnetic radiation travels at the speed of light, we can figure out the frequency of this wave: $c = \lambda f \implies f = 300$ GHz.
We would need an analogue-to-digital converter (ADC) capable of recording 600 billion samples per second, which is outside of what existing technology can do.  ALMA, which observes at exactly these frequencies uses a 4 GHz ADC (see table 1 of an ALMA specification paper).  So how do radio telescopes do it?
The Solution
Lets say you want to observe a 2 GHz bandwidth from 300-302 GHz.
You don't actually care what's going on at radio frequencies less than 300 GHz.
It would save a ton of space to only record the 300-302 GHz channel and throw everything else away.
This can be accomplished with the analogue signal by basebanding it.


*

*Your radio receiver outputs an analogue electrical signal from the radio waves.  This signal is at the actual radio frequencies (RF).

*You run an analogue signal generator at 300 GHz (called a local oscillator, LO) and interfere that with the analogue receiver output.

*The interference produces beats.  The beat frequencies will be at $f_\mathrm{beat} = f_\mathrm{RF} - f_\mathrm{LO}$.

*Now you can record the analogue beat signal with a 4 GHz ADC.


This new beat signal is called a baseband signal.
It contains all of the information from the 300-302 GHz radio wave, but shifted down to the range 0-2 GHz.
If you want to record 16 GHz of bandwidth, you'd need 8 channels each basebanded by a different LO.
In the abstract of the ALMA specification paper it states ALMA can observe 16 GHz of bandwidth.  Going back to table 1, you'll see that it has 8, 2 GHz baseband inputs per antenna.
