Di2 changes screens on 1040 on front of tandem as well as 1030 on back of tandem

On the 1030 just unassign any button action for the Di2 system.

When the 1030 is in communication with the Di2 system go to the sensor details, buttons.

Thanks.  I'll tweak that tomorrow and see if that does it.

So, in the Garmin, while connected to Di2, I see that the D-Fly 1 and 2 channels are connected to the L and R hood buttons.  And, indeed, the default setting seems to be to advance one page when pressed.  So far, so good. The behavior while riding reflects the settings.

But I didn't see any option to "unconnect" the D-Fly channels.  And inside each D-Fly channel options, there's no option that says, in effect, "do nothing".

There is a more global option to disconnect the Garmin entirely from Di2, but I only want to disable the two hood buttons in the rear Garmin.  I'll dig around some more.

If you remove the Di2 sensor from that unit and then add it back it will default to unassigned buttons I think. Otherwise you would need to do a reset of the unit to clear the button assignment. 

I had the same issue when I first implemented Di2 in 2019. While not on a tandem, I ride with 2 GPSs. It was my hope to use the right shifter top button with one GPS and the left shifter top button with the other. I quickly learned that configuring a D-Fly channel with one GPS caused the same reaction on both GPSs. So I concluded this mapping is stored on the Di2 side. (Note this was 4 years ago. My memory may have failed me and/or Shimano/Garmin may have changed something since then.)

So in your case, all I can suggest is that the stoker remove the Di2 as a sensor. Sadly, this means the stoker cannot see things like Di2 battery level or gear choice, but it's better having one's screen change unexpectedly.

Of course all functionally and mapping saved on Di2 side, in battery controller. If it was in another way then it wasn’t necessary to upgrade standard battery to dn110/110A for using buttons on shifters (if wu111 already presents)

To be clear, I have the D-Fly unit (EW-WU111) and I can see and configure the two hood buttons.  I can pick from 5-10 different actions when I click the hood button once, twice or press and hold.  In that sense, its very configurable.

But there doesn't seem to be a way to cause the hood buttons to do nothing on the rear Garmin.  As several have suggested, disabling Di2 as a sensor would stop the problem, but also means the rear unit would not have the benefit of any data from the Di2, such as gear combo or cadence. 

My stoker most often uses the screen with a map plus distance.  So as an interim solution, I've configured the rear Garmin to change to that screen whenever I click a hood button.  That may be somewhat less irritating to the stoker.

But others have suggested that this change in button assignment described above will also occur on the front Garmin (which would be undesirable).  That would make sense if this is dictated by the Di2, not the Garmin.  We'll see when we ride next. 

The Di2 system controls what channel number is sent with the various button pushes. 

It is the Edge that controls what action is performed when it gets that channel number.

The button assignment on each Edge unit is held in non-volatile memory.  Once it has been set per Di2 system on any given Edge it can only be cleared by doing a a unit reset.

Unless I'm missing something, the stoker should still be able to see cadence.

So, after riding for while, I've implemented the interim solution.  The stoker's setting on her 1030 is to always advance to the Map page (same for both buttons).  My setting on the 1040 is to advance forward (or backward) one page.  Since she is on the Map page most of the time, it affects her the least.  She receives cadence and gear combo info.  As someone pointed out, if I don't want to change her screen at all, I can just swipe with my finger on my 1040.  The more permanent solution appears to be this suggestion - a unit reset in which the D-Fly channels on the 1030 do nothing.  I'll try that after we return from a tour and have time to set it all up again.  Thanks for the assessment and advice.