Perhaps another way to look at this issue is to focus on what human nature is, and see what sin changed. Man was created in the "image of God," which certainly included original righteousness (or, as Reformed theologian Louis Berkhof puts it, "true knowledge, righteousness, and holiness"), which was lost in the Fall. Berkhof goes on to contrast this element with others:
But the image of God is not to be restricted to the original knowledge, righteousness, and holiness which was lost by sin, but also includes elements which belong to the natural constitution of man. They are elements which belong to man as man, such as intellectual power, natural affections, and moral freedom. As created in the image of God man has a rational and moral nature, which he did not lose by sin and which he could not lose without ceasing to be man. This part of the image of God has indeed been vitiated by sin, but still remains in man even after his fall in sin.
To put it another way, Adam was a man both before and after his sin. So while the Fall stripped him and his progeny of original righteousness, this is not so essential to his nature that as a result of his sin he was no longer man:
Notice that man even after the fall, irrespective of his spiritual condition, is still represented as the image of God, Gen. 9;6; I Cor. 11:7; Jas. 3:9. The crime of murder owes its enormity to the fact that it is an attack on the image of God. In view of these passages of Scripture it is unwarranted to say that man has completely lost the image of God.
But while one can be said to be in the image of God with or without original righteousness, its lack is a serious impairment:
The loss of that righteousness meant the loss of something that belonged to the very nature of man in its ideal state. Man could lose it and still remain man, but he could not lose it and remain man in the ideal sense of the word. In other words, its loss would really mean a deterioration and impairment of human nature.
On the flip side, this lack of original righteousness, or "original sin," is definitively not an essential aspect of human nature. This is obvious because Adam was a man before he sinned, but it's also clear when we consider the definition of total depravity. It does not imply, as Berkhof and others say, "that every man is as thoroughly depraved as he can possibly become," but rather it is "inherent corruption" that "extends to every part of man's nature."
This distinction makes it clear that total depravity does not mean that man's defining characteristic is his sin – rather, his human nature, as image bearer of God, is corrupted in all its parts. This corruption taints his human nature, but does not eliminate it or replace it.
Obviously Jesus had to have the same human nature as us, but could not have this corruption. So how was he protected from it? By the Holy Spirit:
[The Holy Spirit] sanctified the human nature of Christ in its very inception, and thus kept it free from the pollution of sin.
So Jesus thus has a true human nature, an ideal human nature, different from our own only in that he was protected from the corruption of original sin.
All quotes from Berkhof's Systematic Theology: