Given a (global) number field $K$, thinking of the affine scheme $\mathrm{Spec}\mathcal{O}_K$ can gige an insight into (at least) some kf the number-theoretic terminology, e.g. ramification or local fields. So it seems that this geometric point of view is considered even though, at least in my experience, never explicitly mentioned in algebraic number theory text books.
So when I first encountered infinite primes (places) of a number fields, the first idea about their meaning that came to mind was that those are points "completing" the curve $\mathrm{Spec}\mathcal{O}_K$ the same way an affine curve over a field is completed by several points at infinity. I haven't been able to find any construction that would make this precise, though.
So my question is
Is there a construction that would make naturally the set of all (finite and infinite) primes of a number field into a scheme? Namely, so that the set of all finite primes would form an open subscheme isomorphic to $\mathrm{Spec}\mathcal{O}_K$? And if yes, what are some properties of such schemes? (Are they projective over $\mathbb{Z}$? Are they still normal, i.e. what do the stalks at the infinite places look like?...)
I would appreciate anything from a reference to a textbook where such construction is introduced to a full answer.
I should also mention that I am aware of the Poonen's table and the number field-function field analogy. I am looking for something little more precise than that.
Thanks in advance for any help.