loss of internet connection when using phone

Is your fax line POTS or U-verse Voice?

[If POTS, ] Do you have a DSL filter installed on the jack where the Fax plugs in, or were you professionally installed with a filter in the NID?

fax line is U-verse voice

Correction - my connection is POTS with a DSL filter at the jack

Cool.  Two more questions:

1) is that DSL filter on the phone new, i.e. provided with the U-verse service?

2) Is the U-verse Gateway also plugged through a (newish) DSL filter intended for VDSL2/ADSL2+ equipment?

All equipment provided and installed by ATT. The Gateway is an NVG589. Two telepnone wires run from this Gateway to the wall jack (one from a Gateway jack labled DSL) and a third wire runs from the fax to the wall jack. An additional telephone wire runs from the fax to an office phone.

Unless there is more than one POTS line, normally the U-verse signal and the POTS signal share a line from your home to the local crossbox/VRAD.  The DSL filters screen the frequencies out that each device is not supposed to hear;  The fax (and the phone connected to it) use frequencies in the voice range or about 300 to 3500 Hz.  The DSL modem could use those freqencies, but can't when you're using them for voice, and needs much higher frequencies (that you wouldn't want to hear when making a phone call).

So, there's supposed to be a filter that would filter off the low end for the DSL modem (so it doesn't try to use them) and another that filters off the higher frequencies for your POTS line (your Fax).  Now, the installer could have put a whole house filter in the NID and run separate pairs from that to your wall plate where the DSL gateway and your POTS, but if he had done that, I wouldn't expect a filter on your fax line.

The second line from the Gateway to your wall is likely a backfed U-verse voice.  Do you have U-verse Voice accessible from phone in other rooms in your home?

How long ago was your U-verse service installed?

U-verse voice is available on all other phones in house. Installation was in May 2016.

Okay, your installer did backfeed dial tone as I thought.

At this point I'm assuming that the installer has wired your FAX POTS service and DSL throughout your home on the secondary pair.  The primary pair has been disconnected from the NID (properly) and the connection from the Phone 1 port of your Gateway is providing U-verse Voice service, i.e. dial-tone, for your extensions.

I believe that if your fax line isn't in use when the Gateway trains, and since there is no filter to filter out voice frequencies on it, the Gateway and VRAD decide to use the voice frequencies as part of the set they use for communication.  When the FAX uses the line (or a voice call on the fax line), the conversation disrupts the DSL communication on those same frequencies.

Or it could be when the line goes off hook it creates a large resistance and the DSL signals are reflecting off of it and are interfering.

I asked about when you were installed, because if you were installed in the last 30 days, then your home wiring is covered under an installation warranty.  But we're past that now.  So, rolling a truck to fix something that is technically an inside wiring fault may cost you a truck roll fee.  You might could get AT&T to send you a new two-headed DSL filter and plug the Gateway into the "modem" side of it, but I'm not 100% certain that'll be the end of it.

@my thoughts, how would you advise this customer to proceed?

We do not receive two bills and fax line is same home number as all other phones in house. We had TWC before signing with ATT in May 2016, and there is no modem now except for one installed by ATT.  TWC modem was returned to TWC after switch to ATT. Fax was in place and operating without loss of internet with TWC.  ATT tech arrived to make switch from TWC to ATT. Techs' manager was with tech and did an inspection when tech was finished. Tech service trip scheduled for today and will call manager for post visit inspection if problem isn't corrected.