I know the term "secondary" is frowned upon but I use it to describe myself as quick an easy as I can. As Redpepper's secondary I would not be at all comfortable without having a healthy relationship with her husband.
I had a long term affair at the end of my 16 year marriage and there is a similar energy to how I feel your wife's relationship is shaped. If his wife doesn't know than they are both in an affair..In my opinion, she may be polyamorous but is not in a polyamorous relationship because of the secrecy and dishonesty involved.
On the flip side, I am not polyamorous but am in a polyamorous relationship because of our openness and honesty.
Although I believe there has to be privacy in poly relationships and am comfortable in that, for me, the concept of secrecy or completely “separate lives” is not one that I would be comfortable with. But this is me.
Hope you work your way through this and everyone is healthy and happy