Which holds that each should keep out of each others sphere of thinking/relevance.

A universe with a god will look very different from a universe without
one. Science is concerned with explaining what exists, so the truth
claims of religion are directly in conflict with empirical science. Science is concerned with examining what exists (in whatever form) and
how it works. If one proposes that God exists (in whatever form), then
by definition God falls within the remit of science.

I suppose science and religion could, in theory, be non-overlapping magisteria, but only if
religion withdrew entirely from the universe we all live in. In its
current form, its practitioners can't avoid making statements about the
universe.

Unfortunately, many theists are willing to violate the principles of
NOMA in making bold claims regarding the age of the universe, the
origins of life, the occurance of miracles, etc

Take Christianity as only one example. The bible makes several claims
that have been disproven by science. The universe and everything in it
was not created in 7 days nearly 6,000 years ago, the earth was not
created before the sun, the moon does not cast its own light, there was
no global flood etc.

The Qu'ran has the same stories and other additional claims that proport to tell us scientific 'facts'( see realsom1's post in the politics, culture & religion forum)

IMHO religion does us atheists a great service when it tries to scientifically justify itself, as the cognitive dissonance is deafening and exposes the delusional bubble its adherents are trapped in. It serves to help break the chains that ensnare, via their holy books, the religious believers.

Keep it up I say.

I don't go to mythical places with strange men.
-- Douglas Adams, The Long Dark Tea-Time of the Soul.