I think this question contains half of the answer you are looking for:
as much as mathematics and logic exist in the fabric of the universe; morality and normative ought statements exist in the fabric of the universe.
Then comes a claim I would like to examine:
They are independent of mind, opinions and consciousness.
There is a subtlety here: Math & Logic are developed by us. There are variants of logic besides Classical, Fuzzy and Boolean which exist, and there may be some variants we have not developed yet. The strength of these fields is that they have universal notation, and anyone developing a part of math or logic which comes upon a proof can show-case with great ease.
However, when we get to ethical claims we rely on things such as Kant's Categorical Imperative, or Rawls's Veil of Ignorance. These try to form some formulas for individuals to create an objective morality and law. Rawls addresses the fact that different individuals come from different backgrounds, actually. Human Rights are another go at this, yet it is consensus based. Reflecting upon morality within human-like primates, it seems more like an emergent property which served as the basis for out social norms to begin with.
Positivism is your philosophical doctrine to follow if you want to get in to this deeper: its main claim is that the facts of the world are imposed on the mind by experience. For me, it proves that there is objective sociology, but that it is the least exact, due to it's complexity. Once the third stage of spiritual development is achieved, all of humanity would be perfectly moral and live in eternal peace. However, religions and cults do exist. Many people simply live their lives holding cultural values close to heart, due to the fact that they truly did grow up in their environments. Even with the positivity stage being just, true and it is really the final stage of social evolution, the road for it's full realization hasn't been structured on a universal level. There are, factually as you experience in your mind, beings who hold entirely different moral values. Islam calls for Jihad, whether you believe in the Quran or not. And some Muslims believe in this. The Torah calls for the Jewish people to reside in Israel. And some Jews and Christians believe in this.
Kant, at least for me, proved that reality is structured by the mind through some categories which are necessary and sufficient for our understanding and scientizing it: All STEM depends on our ability to learn from others alongside our creativity and utilizing the tools we acquire. Kant also proved that antinomies, such as the debate about beings such as gods, can continue eternally without resolution. They are undecidable by nature, and we might never develop a narrative in all of society which will impose itself as true, objectively, although I do hope we will.
One may be a realist, and still see morality as mind-dependent.. and evolution-dependent.
Hope I answered your question, I felt ought to try!