There have been very good answers, but I want to add a few thoughts.

By any sane definition (sorry, Hobbes) of the terms, theism is incompatible with materialism. But the reverse is not true, substance dualism and atheism are compatible.

Atheistic dualism is even a common view in the world, because of Buddhists and Taoists.

But if we look just at Westerners, we get a different picture. So we must ask ourselves: What (in the West – for now on) makes (substance) dualism and atheism seem so incompatible? Why do atheism and materialism seem to go hand in hand? Why are nearly all prominent atheists also materialists?

And why do discussions about consciousness get so heated? Just look at the hundreds of angry comments to an article from atheist philosopher Thomas Nagel, a summary of his book which attacks materialism (he doesn't commit to substance dualism, though).

Substance dualism as defined by you (the OP) doesn't guarantee even the one important thing, which would be persistence of the self after death.

Superficially it would not change much, except our attitude to free will, perhaps. So, why again, do atheists take it so serious?

IMHO humans seem to suffer from an intense obsession to construct a world view, which can give us a satisfying cosmology, i.e. which can give a good answer to ‘The One Question’, the question of “arche” (ἀρχή):

“What is the ultimate principle that underlies the universe?”

And materialism does this. It answer this question very elegantly and convincingly (which does not mean correctly). The ultimate principle is objectively observable, purposeless ‘physical stuff’. The universe is inherently meaningless, there is no goal or point to it. It just exists. And by operations of chance and necessity everything else has come to be.

While we may dislike this answer, it allays our thirst for further probing deep metaphysical questions. For example, “Why exists something rather than nothing?” – what's the point of asking that if what exists is just inherently meaningless? Or, “What is the meaning of life?” - either there is none or the meaning we personally give it, it is not tied to the cosmos.

Theism, of course, is also a world view which at least gives us the illusion to answer ‘The One Question’.

But what about atheistic dualism (or atheistic non-materialism)? It fails to answer ‘The One Question’ satisfyingly. In substance dualism we must just take it as a brute fact that two fundamentally different substances exist. Now, why does ‘mind stuff’ exist? Why? Where does it come from? A materialistically conceived evolution surely can't explain it. How can we dare to believe ‘consciousness’ is equally deeply built into the universe as fundamental physical properties, like say, electrical charge or mass?

Outside of a theistic context it really sounds like the universe from *cough* not that intellectually challenging science fiction. Like Star Wars... or Warhammer 40.000.

Why does such an extremely weird universe exist? This is beyond “quantum mechanics” weirdness, it is unacceptable weirdness.

“Why is there something rather than nothing?” definitely becomes a pressing answer again. Also “What is the meaning of life?”: there is no transcendent God – but there seems to be a profound connection between human life and the natural order, since consciousness is a fundamental part of the fabric of the universe. Maybe there is a cosmic meaning? But on the other hand, we are not living in a universe that is very nice to its creatures. Even if we live in the best circumstances, there still is death and suffering all around us, psychologically healthy humans are just able to ignore it. As Goethe's Werther puts it:

No: it is not the great and rare calamities of the world, the floods which sweep away whole villages, the earthquakes which swallow up our towns, that affect me. My heart is wasted by the thought of that destructive power which lies concealed in every part of universal nature. Nature has formed nothing that does not consume itself, and every object near it: so that, surrounded by earth and air, and all the active powers, I wander on my way with aching heart; and the universe is to me a fearful monster, for ever devouring its own offspring.

So atheistic dualism (if we don't incorporate more extreme ideas from Eastern religions) is, at least in my view, a very unsatisfying, weird and lopsided world view. That doesn't mean it is incorrect, of course.

As a final note, the truth of materialism is unsurprisingly part of the secular elite's orthodoxy. Probably most of them want to live in a climate in which religion/theism is regarded as intellectually not respectable. If dualism (or some other non-materialism) would become plausible again, their whole narrative would begin to unravel.

This is also a result of not being content enough to just use the classical argumentation against theism, but to present a complete alternative world view, in which there is not even a foothold for religion, and to use the prestige of science to advance it by framing the dispute as a fight between scientific rationality and archaic superstition.

If someone felt so much to be on the right side of history and arrogantly presented himself as the voice of reason, to turn out being wrong is a death blow. In light of this, the strong hostility of the secular elite to criticism of materialism is not surprising at all.