I am in much the same boat as you. Wayward wife, in some sort of affair, definitely emotional, POSSIBLY even probably physical (though can't definitively confirm beyond them having spent night together)... I have some ability to monitor her but am trying to not become obsessed with doing so. She gave me a "no contact" promise but has not yet provided me her email and social media passwords ("I don't even know what they are anymore, the apps just pop up on my phone"... "What can I possibly do wrong on facebook"... etc etc.). I probably made a mistake in not establishing transparency as a "boundary" when I did likewise for "no open marriage" and "no contact", but, either way, looks like we are facing a bit of the same sitch, there.
My take is not quite as urgent on this: Yes, it would be a nice sign if she would provide this (and I brought it up with mine last night, opened the door, and will bring it up again in a couple of days), but, at the end of the day, in this day and age there are about 10,000 different ways that a WW can contact her AP surreptitiously. (Create new email or other accounts, get a "straight talk" or other pay as you go phone using cash, even installing vonage on their phone so calls don't get registered with your provider. (And, if they're really slick, they can hide apps in the background so they are not easy to find in a cursory search.) If she's gonna cheat she's gonna cheat. My take has been to try to snoop/monitor in a measured way (although it is a bit easier with my W than perhaps with others because she is a truly inept liar and secret-keeper) while encouraging her towards more openness. I did get her to give up the "cheater phone" and she does leave her own phone lying around from time to time "unguarded". I told her last night when we were talking that she knows what I won't accept in our marriage (the boundaries), that I was thankful she had agreed to cut contact and do the transparency stuff she agreed to do, but that I thought the additional openness would be helpful to her and to us as we strive to rebuild trust. Remains to be seen if that resonates with her. But, honestly, and others may disagree here, the failure to turn over all passwords, etc., doesn't seem like that huge a deal. A true WW, IMO, might easily agree to do that and then just go out and get another phone and/or open up a bunch of new online accounts. I am going to push it a bit with mine but I did NOT establish that (passwords) as a "boundary." If she slips up, however, and has contact with the OM, any future attempts at reconcilliation (after I have done whatever I have to do to separate us) will darned sure include it.
My understanding is that with WWs who are just "giving up" an A and an OM will be in a surly sort of grieving process. I am only 9 days into this with mine, so I am sort of taking baby steps for a few more days, I think, until I see how it's going, and I did not establish full internet "transparency" (though I did insist the cheater phone go bye-bye) as a boundary. Now, if you did so, you may have to be more strident because once you establish a boundary you need to be firm in enforcing it or they all become somewhat meaningless.
Don't know how helpful any of this is, but thought I'd offer it up. My own sitch has some other weird/troubling overlays that complicate things, and i think I am in a somewhat tighter "spot" than you, but we shall see.
Remember that you can't control what she does, you can only control you. Don't obsess too much on the monitoring/snooping-- it can become consuming. Get out, GAL, be good to yourself.
Thoughts and prayers to you and yours.
H53/W51, R-ing 4/'18
"Do not arouse or awaken love until it so desires"-Sg.of Sg 2:7
"So oftentimes it happens,that we live our lives in chains, & we never even know we have the key"-Eagles III 1:3
