Ok. Well then. First off, you can never give anyone "objective evidence" since objective is always external to our minds, and our experience and how we take in information and knowledge is ultimately subjective. It's like asking, "give me a square circle."
Fair enough. Actually, far better than that. I see this as monumental concession. But let's go on.
Secondly, the way we can assume there's an objective existence is that the same problem applies to both reality, nature, and God. If God exists, then there's existence that's objective to us.
According to what you said above, the objective would be information and knowledge that is ultimately subjective. Feel like that is highly pertinent going forward. Perhaps not, and if there is a dispute on this, I'm sure I'll hear about it.
If God doesn't exist, nature still can exist, and if nature exists, it will be something that exists objectively to us. Now, if we assume the opposite, that nature and God are all subjective in existence, well, then your world is your world and there's nothing else, then why ask or bother try to learn anything about it? After all, if all that exists is only your subjective construct, then whatever you want to be true is true in your world. Your mind is then your master, and there's no God besides yourself. So, either we assume nothing is objective, and the world is only in your mind, or there is something that exists that is objective, and nature is by default it, and God could also be part of that objective existence.
Even in this rhetoric, I see you making assumptions. Pretty sure you don't deny that, but kind of feel like you might dispute certain assumptions. I welcome that. I see the assumptions being based about objectivity as faith-based. Thus, I argue that physical existence 'exists' because of faith in it. Not because we have 'reason to understand it is (in fact) objective and (is actually) independent of the mind' but because it is what we choose to hold faith in.
Some of what you are writing in the paragraph is conceptions I've heard many times before, but seem to be relegating existence down to a solitary perspective, when my experience along with my faith in the physical would strongly suggest it is intersubjective. IOW, I don't believe, nor observe that I am doing this alone, but acknowledge it might be working that way. I just don't make the additional assumptions that you have in this paragraph, "that I am the only God that exists," and that only my (human) perceptual subjectivity exists. I have to stipulate this stuff because there is a key part of my understanding that isn't far off from what you are purporting, but I do understand that in reality, you are me, I am you, and the diversity of our beings is actual, even while it is in no way separate (in reality). So "no God besides myself" implies to me, in way you are using it, that my 'self' is only my human construct about 'me' whereas I see my Self as 'us' without exclusion.
I'm with you on the nature is by default the connection, or what you call objective, and what I'd call intersubjective (though I'm really not too hung up on what we call it).
Thirdly, natural science is all about empirical testing of nature. The assumption is that there is an objective world to be tested, it might not exist and be just subjective mind-constructs, but it's goal isn't to prove that but to test the world that is assumed to be the object of the test.
IMO, the assumption preceding this and pertains to the subjective in a quasi-real way is that I self identify as being in a physical body, or even as if I am that body. This is where I generally ask for the evidence that I seek. How could I provide objective evidence for existence of the physical world without relying on perception, which is by default saying, "I accept that I am a physical being, and have physical senses, therefore let me now provide the evidence that is independent of my body (and mind) that shows the nature of 'physical' reality."
To me, you've already conceded that information/knowledge about me (or anyone) being a physical being is ultimately subjective. That there is no independent way to verify that, and that it is an assumption, or what I would call a matter of faith.
I would concede that once that faith is accepted, then 'natural science' would allow for the 'objective' world to be tested, as if its existence is now reality, when actually it is taken for granted, based on a (monumental) faith proposition.
And lastly, so from that, why do anyone have to prove to you that objective existence exist or not? Or that reality is real? The goal of natural science isn't to prove God, something even beyond objective reality, but to test this world.
Same reason for having to prove objective existence as to why ask for an argument for God if conceded that ultimately everything is subjective? If there is faith in objective/intersubjectivity and doubts about nature, God, physical phenomenon, then perhaps intellectual reasoning could help dispel doubts about certain conclusions that are reached which (arguably) affect everyone. The goals of spirituality, divinity do not seek to disprove scientific understandings. I see them as perfectly allowing for all of that (and more). But like all things debate oriented, I'm sure each of us can find persons who think the existence of one disproves the other and therefore the other is without any credibility and all its adherents ought to be ignored or punished in some way that makes us feel even more righteous.
Why do we need to? What is it that you're looking for? Are you suggesting that God is more likely to exist because the objective reality is most likely subjective?
You've already conceded on the last question. If you feel you haven't, then I still welcome whatever you wish to provide as objective evidence for the reality of a physical universe.
As much as I've written in this post, this just barely scratches the surface of where I could've gone, have gone, and am still learning about myself. The argument for God which I presented embraces intersubjectivity or what could plausibly be called objectivity, but does speak directly to concept of 'self' as if there is very pertinent information at all times within the physical universe (and where consciousness is present) to understand 'what is this for' and 'Who is making the observations' that ultimately reinforce beliefs about the nature of reality, self and (if you will) God.