Science can never be assumed to have tried as hard as it can; it's always possible that some new information will come along and explain something that wasn't explained before. So if you're thinking scientifically, the appropriate notion isn't "inexplicable" but "not explained as yet".
Anyone -- dualist or otherwise -- who says "OK, we've done enough now; time to give up" is engaged in thoroughly unscientific thinking. Since scientific thinking demonstrably works well, that's probably a bad thing.
(The above leaves some space for a sort of provisional dualism. You could say "So far, no physical explanation of consciousness is apparent. So we might as well treat it as an independent thing until such time as a physical explanation comes along." That would be fine if we had no information at all about the relationship between consciousness and matter. But in fact we do have some information. We have evidence that particular aspects of consciousness are related to particular bits of the brain and particular things the brain does. We have partial explanations, where some things about consciousness have somewhat-handwavy explanations in physical terms. And we have a long history of trying out non-physicalist hypotheses about things -- gods and ghosts and vital spirits and so forth -- and finding them, again and again, smashed to bits once knowledge advances far enough for them to come into contact with reality.)
Science can never be assumed to have tried as hard as it can; it's always possible that some new information will come along and explain something that wasn't explained before.
So were the cases where, as a matter of fact, science has posited a new property or force (spin, colour charge, etc) illegitimate?
We have evidence that particular aspects of consciousness are related to particular bits of the brain
We have evidence that the N fundamentals of physics nonetheless interrelate.
ETA
...And we have a long history of trying out non-physicalist hypothes
Background
I was raised in the Churches of Christ and my family is all very serious about Christianity. About 3 years ago, I started to ask some hard questions, and the answers from other Christians were very unsatisfying. I used to believe that the Bible was, you know, inspired by a loving God, but its endorsement of genocide, the abuse of slaves, and the mistreatment of women and children really started to bother me.
I set out to study these issues as much as I could. I stayed up past midnight for weeks reading what Christians have to say, and this process triggered a real crisis of faith. What started out as a search for answers on Biblical genocide led me to places like commonsenseatheism.com. I learned that the Bible has serious credibility problems on lots of issues that no one ever told me about. Wow.
My Question
Now I'm pretty sure that the God of the Bible is man-made and Jesus of Nazareth was probably a failed prophet, but I don't have good reasons to reject the supernatural all together. I'm working through the sequences, but this process is slow. I will probably struggle with this question for months, maybe longer.
Excluding the Supernatural was interesting, but it left me wanting a more thorough explanation. Where do you think I should go from here? Should I just continue reading the sequences, and re-read them until the ideas gel? I'm coming from 30 years of Sunday School level thinking. It's not like I grew up with words like "epistemology" and "epiphenomenalism". If there is no supernatural, and I can be confident about that, I will need to re-evaluate a lot of things. My worldview is up for grabs.