So what exactly is going on, you make a phone call and try to browse the web/use data and no data? When this happens the phone is using 3g for the phone call correct? Is the phone unlocked from an apple store, or bought locked from a Rogers store? When you goto: settings -> about: carrier version Rogers 27.1 ?
Yes i believe the user has the same issue as what the OP is having and title is saying ( as their original post saying it). Even if it's unlocked by Apple it still should work as a unlocked model shouldn't have a problem with this. I have had a couple unlocked iPhones and worked.
This seems to be a issue with the network or software causing this. A lot of issues like this have been reported in the Apple forums and got told to go visit the Apple store and exchange the unit or contact their provider and enable it on their side which the user has done already.
Best thing to do is for them to head to the Apple store and her troubleshoot the problem with them. They should be able to assist them more and swap the decide out if needed be.
When I'm not on call I have LTE and data works fine but when I call it goes to 3G while I'm talking and want to access any app or google no data at all this is iPhone 6 I bought it from bell 2 years ago and it is unlocked after that went with Telus and was working good i speak with Rogers Rep and they walked me through to reset the network but didn't help
The device should work no matter even if it's been unlocked by another carrier as Rogers does not block this.
Once again I would suggest either factory reseting the device to see if that will fix the issue OR book a appointment with the closest Apple store and see what they can do. That's the only two things can do as Rogers has already did their best to help you. Apple will diagnose the issue and they can either repair it or most of the time exchange the unit out for you. They will put the sim card in and make sure all works before you leave.