5.56 vs a .223 chamber is not the issue. The Tula ammo is loaded to .223 spec anyway. The expansion/contraction rate of the steel cases differs from that of brass, making the ejection process more sensitive to timing. Having a chrome lined chamber does help because it's slicker.
The two most common problems with shooting the steel cased stuff are stuck cases and short stroking and I don't think I've ever seen a case of a rifle suffering from both. In the case of the short stroking, if could be because of gas leaks, less that ideal gas rings or the rifle being overgassed just enough to slow the cycling process down to a point where the bolt just doesn't have the momentum to toss the spent case.
I haven't seen it all, but in the rifles that I've worked on that were suffering from stuck cases, it was because of overgassing. The gas port in the barrel being drilled larger than spec. In an overgassed rilfe, the bolt is unlocking to soon and trying to eject a casing that hasn't contracted yet. The easiest thing to try is to slow the ejection process down a bit by adding weight to the ejection system. Try a heavier buffer. If you have a semi auto bolt carrier, try a full auto bolt carrier. In extreme cases, I've been able to get rifles running by using a clamp on gas block and choking off some of the gas. Personally, if I put something together that gives me this sort of trouble, I'll yank the barrel and get another. It's easier than trying to diagnose the problem and potentially cheaper in the end.
Shooting steel cased ammo is a great economical way to enjoy your AR. Unfortunately, some manufacturers do over gas their rifles to insure reliability with brass cased ammo, which a bit of overgassing can do.
What make is the upper? When you shot the Remington ammo out of it, what was the ejection pattern like? Was it ejecting the brass forward at the 1:00 position. Brass being thrown forward is generally an indication that the rifle is over gassed. If you have ejection at around the 3:00 position, that's perfect and you shouldn't be having problems with the steel cases ammo.
I subscribe to the school of thought that an AR that won't run on all ammo has something wrong with it, but if your friend is willing to live with it, I'd try some Silver Bear.