Phone reception puzzle
I had that problem when I was in the UK using a giffgaff SIM on my Moto (Marshmallow). My daughter's phone was getting a signal when I couldn't get one bar. Checking my APN settings to ensure they were as giffgaff intended solved the problem.

Assuming you both have exactly the same set up otherwise, if the APN setting is different or non-existent on one of the phones, your wife's phone may be trying to connect to the wrong network (or, if she has roaming disabled, she may not have a network to connect to).

First guess? Check APN settings are what giffgaff gave as the settings. It may be that hers are different from yours because for whatever reason giffgaff had the SIM cards set to use different networks. So check the text which provided that information on her phone as well as yours.