How does Constructive Quantum Field Theory work?

Please correct me if I'm wrong, but it seems to me that two and three dimensional axiomatic quantum field theory were constructed as follow: the wightman axioms were formulated in euclidean space via a wick rotation, and the theories were constructed in euclidean space rather than Minkowski space. Now, I have two questions... first, were the euclidean theories then wick rotated back into Minkowski space? Otherwise, how else were they converted to Minkowski spacetime? Also, because of the introduction of a complex coordinate, an n dimensional Minkowski qft would have to be formulated in an n+1 dimensional euclidean space, right? If so, is it the addition of a coordinate that makes it difficult to formulate 4 dimensional qfts using constructive methods? Or does the addition of a coordinate actually help in some convoluted way? Sorry if the questions are vague; I am still introducing myself to mathematical physics. Also, any level of answer is fine, although I would prefer technical to layman level.

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