Interplay between the cosmological constant and “microscopic” properties of string vacua

As far as I understand, string phenomenology is usually concerned with compactifications of string theory, M-theory or F-theory in which the uncompactified dimensions form a 4-dimensional Minkowski spacetime. However, we know our actual universe has a positive cosmological constant hence its asymptotics are that of a De Sitter spacetime. On the intuitive level it makes sense to me, since the microscopic physics should have little to do with spacetime asymptotics. However, from another point of view I see a problem.

It seems to me that a cosmological constant in the effective 4-dimensional field theory requires a non-vanishing Ricci tensor in the compactified dimensions. For example, the classical case study for anti-De Sitter string theory is AdS_4 x S_6. The compactified dimensions form the sphere, a manifold with positive curvature, compensating the negative curvature of AdS.

This non-vanishing Ricci tensor seems to require different topology from a vanishing Ricci tensor. Hence all standard compactifications like Calabi-Yau manifolds, G2 manifolds etc. don't seem to be compatible with a non-vanishing cosmological constant.

What am I missing here?

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First of all, in the most recent decade, string phenomenology isn't talking about strictly Minkowski vacua. E.g. in the KKLT paper, you will see $AdS_4$ vacua uplifted to $dS_4$ by antibranes and no Minkowski space at any place in between.
The fact that a nonzero C.C. is generated for the large 3+1 dimensions doesn't mean that one can't find any shape of the hidden dimensions that exactly obey the equations of motion. Just like there exists a "tiny C.C." deformation of the flat Minkowski space, namely the $dS_4$ space with a small C.C. around us, there also exist solutions for the compact 6/7 dimensions that have a tiny (but nonzero) Ricci tensor proportional to the Ricci scalar. In the Calabi-Yau case, these deformed solutions will strictly no longer be $SU(3)$ holonomy manifolds; they will be $U(3)$ holonomy (Kähler) manifolds if we acknowledge that the Ricci curvature, while tiny, is nonzero.