Tullian Tchvidijian's vocabulary, if not his theology, is definitely not in line with the typical understanding of total depravity. We'll begin by explaining that, and then comparing Tchvidjian's teaching.

Total Depravity

Robert Reymond's definition of total depravity is typical:

Both because of original sin and their own acts of sin, all mankind, excepting Christ, in their natural state are thoroughly corrupt and completely evil, though they are restrained from living out their corruptness in its fullness by the instrumentatlities of God's common grace. Accordingly they are completely incapable of saving themselves.

The phrase "in their natural state" is worth noting here, as are the references to two related concepts: radical or total corruption and complete inability. R. C. Sproul includes both in his treatment of total depravity as well:

To be totally depraved is to suffer from corruption that pervades the whole person. (118)

The moral inability of fallen man is the core concept of the doctrine of total depravity (128)

Sproul also points to the Westminster Confession of Faith, which makes the same points about the condition of fallen man in 9.3:

Man, by his fall into a state of sin, hath wholly lost all ability of will to any spiritual good accompanying salvation: so as, a natural man, being altogether averse from that good, and dead in sin, is not able, by his own strength, to convert himself, or to prepare himself thereunto.

Tchividjian

With this understanding of total depravity, we can look at Tchvidjian's views. He is clearly correct that the phrase means more than one thing; it does indeed include both total corruption and complete inability. And he accepts, naturally, that the complete inability aspect does not apply to Christians, as they have been made alive in Christ.

So what about the other aspect? Do Calvinists believe that Christians are still totally corrupt, that is, corrupt in their whole person? This part is a bit trickier. Tchvidjian writes:

Even after God saves us, there is no part of us that becomes sin free–we remain sinful and imperfect in all of our capacities, in the "totality" of our being.

Charles Hodge, in distinguishing three types of "good works," agrees: "not even the works of the holiest of God’s people are good" when we consider the phrase to mean that both the action itself, as well as the motives of the actor, are completely in line with the law. However, he continues:

The works of the children of God, therefore, although stained by sin, are truly and properly good, because, (1.) They are, as to their nature or the thing done, commanded by God. (2.) Because, as to the motive, they are the fruits, not merely of right moral feeling, but of religious feeling, i.e., of love to God; and (3.) Because they are performed with the purpose of complying with his will, of honouring Christ and of promoting the interests of his kingdom.

So in one sense, Reformed theologians agree with Tchvidijian – sin does continue to taint everything that Christians do. But it's inaccurate to say that sin taints everything that Christians do in the same way that it affects everything that non-Christians do. Hodge makes a clear distinction between the "civil goodness" of the unregenerate – the "good" works done as a result of fear of punishment or social pressure – and the "good works" of the Christian, which are performed out of a love for God and a desire to honor him. Tchividjian fails to make this distinction, and in doing so fails to accurately portray the teaching of total depravity.

Summary

Tchividjian's divergence from traditional Reformed theology appears to be more a matter of vocabulary and lack of clarity than actual theological disagreement. The two points of discrepancy are:

• He uses a label that means two distinct things as if it may mean only one thing. The doctrine of total depravity includes both total corruption and complete inability, but though Tchividjian admits that complete inability does not pertain to Christians, he still applies the total depravity label to them. From a traditional perspective, at best he can say that only half of total depravity applies to Christians.
• He misuses the label of total corruption. As a doctrinal label, total corruption denotes a heart that does nothing out of love for God, but with such a definition it cannot be applied to Christians. Instead, Tchividjian confounds this with the idea that all actions of Christians are stained by sin, which Reformed theologians agree with, but call by a different name, such as remaining sin.

References

• Hodge, ST, III, 18.4
• Reymond, A New Systematic Theology of the Christian Faith, Appendix E
• Sproul, What is Reformed Theology? Chapter 6