There are no Carnal Christians. However, we all are carnal Christians.

The Campus Crusade for Christ  (rebranded as "Cru") was (is?) very big on the idea of a Carnal Christian.  (Let it be clear that while I disagree with Cru on some theological issues, I have great admiration for the organization and our differences are not foundational.)  They liked to use a series of images depicting a person's spiritual state using a circle and a throne. Christ, represented by a cross, could be outside (unbeliever) or inside (Christian) the circle. The person, represented by the letter 'S' (for Self) could be on the throne or Christ (the cross) could be on the throne. One of the permutations, then, was:

This was (is?) the CCC's representation of a Carnal Christian. The person has accepted Christ, but Christ is not on their throne. Christ is their savior, but not their Lord. The problem for them is that no such person exists. The people in this state are not Carnal Christians. They are simply unbelievers masquerading (probably through easy-beliefism) as Christians. One who puts himself squarely on the throne seat is not a Christian.

A so called Carnal Christian is in any case a heretic. The heresy is antinomianism: the belief that if we are free of the law then we can ignore the law.

Free from the law 
Oh blessed condition
I can sin all I want 
And still have remission

Paul addresses the heresy full force in Romans, summarized nicely by Rom. 6:1-2.

The book of James also utterly refutes the very concept of a Carnal Christian. If you are saved, then your faith is not dead. But James writes that a faith without works is a dead faith. Ergo, a Carnal Christian, by definition without good works (except, possibly, for imitations) has a dead faith. So a Carnal Christian is not saved and therefore is not a Christian at all.

It is impossible to be regenerated by the Spirit and not have your life changed. It may not be outwardly radical, and it may be slow, but your life will change. James offers no probationary period where you are excused from works yet your faith is alive.

However, a Carnal Christian is an approximation, at times a very good one, to a lower case 'c' carnal Christian.

The phrase "carnal Christian" is hard on the ears. The definition I will use is that a carnal Christian is one who accepts Christ but struggles to conform to Him. The key word here is struggles.  The mythical Carnal Christian does not struggle. The carnal Christian does.  The picture of a carnal Christian would have the cross on the throne, but the 'S' would be rocking the chair in an attempt to displace the cross, or the 'S' wold be trying to climb up on the chair.

A carnal Christian does not mean the person is "as bad as can be" and is engaged in near constant immorality. It can come in a nice family-man package. A carnal Christian can go to church; he can be a deacon, elder, pastor or priest.

A carnal Christian might ace a theology exam. But he displays less fruit than you would expect, yet there is not a complete absence of fruit. A carnal Christian is one who fights a constant battle between conflicting desires to please God and to seek the pleasures of the world. You know many such persons, namely because the set of carnal Christians is the same as the set of all Christians. Paul spoke crystal clearly on this matter in Rom. 7:14-19.

There are no Carnal Christians. However, we all are carnal Christians.