Okay, now I am not going to be nice and I will call you out. You don't have any version of Objective Authority over me as for your "we" and I don't believe in your religion. So stop claiming authority to speak for a "we", that is not there and stop considering your individual belief system as true for all humans.
So how is "reality is real" a religious claim? Well, it is simple. It is a supernatural claim, because it is a claim about the universe or everyday world, which is not observable as a part of the universe, yet considered true or a fact of the universe. I.e. it is: Of or relating to an order of existence beyond the visible observable universe.
How? Well, I can't see reality nor can I see real. "Reality is real" is not different than "God is real". All 3 words have no observable properties and thus objective referent. None of them are a result of external sensory experience and both reality and God are used to judge other humans by some humans.
How? Well, you claim that you speak for all normal humans in that I must believe in "reality is real" like you do or I am in a closed facility.
So let us make a reductio ad absurdum on that. I am sitting in my rubber cell totally catatonic and I have been so for over 20 years. In that time I imagine I am on the Internet part of the time and as a skeptic, I have been told, that I am not really in contact with reality, because those in reality, keep telling me that I can't doubt that "reality is real". But I can. I doubt that just as I doubt other religious claim and they, including you, keep telling me, that I am not in contact with reality.
So here is the joke: How can you as group of believers in "reality is real" and you as a part of reality in reality, keep telling me, when I am not in reality, when you do that from reality? How do you as in reality communicate with me, when I am not in reality?
Yeah, you as an individual are in effect a dogmatic fundamentalist, because we are not allowed to doubt "reality is real" and figure out that it is a supernatural claim, which is not true of the everyday world.
In effect you apparently as an individual only accept doubt that confirms your subjective, individual and in effect religious world view. At least for now.
So back to the OP. Yes, we should doubt any religious reasoning and that includes your "reality is real". I was taught too well and have learned too well to doubt any claim, that I treat your sub-culture of in effect rational skepticism as just another belief system and your in effect subjective insistence to this effect: "... But axioms are often called "intuitively true" and unnecessary to prove for a reason. ..." That is all well and good. The problem is that there are no objective, universal reasons, because all reasons are in effect temporal, local and subjective.
I.e. that you with your brain make sense in a certain way as with reasons, doesn't mean, I can't do it differently.
So yes, again. We ought to doubt all religious reasoning and that includes yours.
So here is the reason that I go off on you. I am a "card carrying member" of the sub-group humans, you use as a moral/evaluative negative. I am just so lucky, that my medication works, that I have a wife, that takes care of me and that a part of my brain can compensate for the part that is crazy. But if that was not the case, I would probably in effect be one of those you called out.
Now as crazy I have had to learn that I can't be like you and that in effect means that I believe differently than you. Yet I am still apart of the everyday world, though I don't believe "reality is real" like you do.
So now you can report me, if you have to. But in effect you belittle a group of humans and made them in effect a "non-we". But you know what? I don't need your "we". I am doing good enough as me and I don't need your subjective reasons for how you make sense of the everyday world. I can do that on my own.
So as a religious person I have to do, what I always have to do for most debates. Call out other religious humans for claiming an authority, they don't have. And that includes you.
Mikkel