I'm a system integrator, an installer for video production systems. I cater mostly to town halls and cable access. During the "bat flu" most towns resorted to Zoom, Microsoft Teams or other virtual meeting software for their public town meetings. Once everyone returned to the office, it was back to the installed production systems—only now everyone wants to fold the virtual meeting (I'll call it VM) into the works.

This sort of "hybrid" meeting creates the same problems once encountered by radio broadcasters when taking a phone caller live on the air. The telephone is a full-duplex system, with two send/receive channels in use. Telephone users can interrupt each other. But broadcast radio is half-duplex (or less) with only one send/receive channel, and used only one-way. To bridge the two systems required a box known as a "hybrid."

In audio and video production, there is already a technique known as "mix-minus" that allows the withholding of selected audio to avoid feedback—namely, suppressing microphones from the return feed to the studio. But how do you "mix-minus" with video? If you send the same PGM video to the VM participants that goes to the cable TV audience, it will produce "video feedback"—a hall-of-mirrors effect like the Doctor Who titles, or one of Nam June Paik's creations.

(If you are using vMix's Zoom plugin, all of this is fixed already. But not everyone uses Zoom for an installer's convenience.)

There are numerous approaches to the problem. Some simply keep the two systems separate. VM participants appear on a big screen for the local audience. And cable TV audiences have to suffice with a camera pointed at the screen open-air. This is not an ideal arrangement, especially if a text document is being "screen shared" among the VM users. I once assembled a two-switcher system for one client. Although instead of switching two shows in parallel, I set up one switcher as the "primary," which cut between all the various sources, except the VM source. The primary PGM was fed to the VM participants. Another copy of the PGM fed a secondary switcher to the cable TV audience, only it was a simpler A/B switch, cutting between the primary and the VM source whenever the cable TV audience needed to see them.

I've also beat my head on the possibility of using macros and scripts, which a system like vMix can do. When suddenly a simpler approach occurred to me, an approach almost any video switcher can do. And I was too wrapped up in cool, high-tech solutions to realize it before now:

Use the VM source strictly as a full-screen overlay, like a title. Never cut to it or take the VM directly. If the return feed to the VM audience is a "clean feed," which most switchers can do, they get the bonus of a fully intercut show, but never video feedback.

Some users may have more exotic "two output" requirements. They'll most likely have to resort to the two switcher method (or use a multi-M/E switcher like Blackmagic's ATEM). But for all my usual clients, this overlay and clean feed technique is easier, simpler, and less confusing to use.

I know the effect happens, as I have tried it—or rather, been forced into it. (I knew what would happen.) I know vMix and OBS are designed to have a single output, recorded and streamed. But as described in the OP, I've been forced into finding a solution for a mixed audience.

If vMix were used like a multi-sourced "Webcam" for a virtual meeting, there would be no problem, as every participant already has the benefit of seeing others in the system, as well as "screen-shared" documents. The hitch is the cable TV audience—they need to see the virtual end of things. And that produces the feedback, unless you come up with some way to "mix-minus" the video return to the virtual side.

Many video switchers have "auxiliary" outputs that can be individually switched. In fact, vMix has this very feature. But using it to avoid the feedback means independently switching for each audience. And the vMix "outputs" would be limited to cuts-only. The clients I work for will not settle for one, wide-screen view of the room to their virtual audience. They want the fully switched experience.

Ergo, the simple, "automatic" way of satisfying both audiences is the "clean feed" approach I described. In that regard, vMix has an advantage in selectively turning off only one "overlay" to the virtual side. All other overlays can be used. If the virtual meeting is always overlaid for the sake of the cable TV audience, and that overlay channel is excluded in the virtual return, you get a video "mix-minus."

Perhaps the difficulty in seeing this problem is if you come from the Internet streaming side of things. I'm dealing with clients who already have production systems installed to 'cast live meetings to a cable TV audience. Virtual meetings are the new kid. The clients want that included. And that's where the problem comes in.

vMix is certainly more sophisticated than most other video switchers on the market in its price range. I have the 4K version, and yes the Output input in the upper tier versions could do the trick. I'm still fairly new to vMix and learning its features. I'm very impressed with the customization possible. Anyway, the overlay/clean feed approach I describe would work with the lesser versions via Fullscreen output, correct? (As well as any other switcher on the market with a clean feed output, which includes many of my clients who are not using vMix from the start.)

As for "hiding my own feed in the meeting software," that is not the problem. Obviously, I don't need to see myself. But when a remote participant is speaking, that is exactly when you are going to select them as a source. Your cable TV audience wants to see them. If you send the same thing back to the virtual meeting, you get the video feedback. Hence the "mix-minus" approach with overlay/clean feed.