In the semialgebraic context, Delfs and Knebusch defined in 1985 their "locally semialgebraic spaces" and later (only Knebusch) "weakly semialgebraic spaces" as some infinite gluings of semialgebraic spaces. But the majority of model theory seems to be carried out in Mn, where (M,...) is a structure (a kind of "affine" situation).
Do model theorists need to pass to infinite gluings from time to time?
No, so far is glueing of spaces not a general technique of model theory. In the cited situation, the question was if some knowledge or methodology from differential geometry and algebraic topology are reflected by the first order theory of the ordered field of the reals such that those knowledge and methodology can be transfered to real closed fields, inspite of the fact that the general real closed field lacks the topological conditions to do differential geometry and algebraic topology, not being locally compact or locally connected. The most natural answer is to imitate the differential geometry so well one can, and to substitute the word "smooth" with notions like "continuous and definable", "definably differential" (etc), where definable means here "semialgebraic".
Similar constructions were done in the most general context of O-minimal theories, differential theories, exponential fields, pseudo-finite fields, and in some combinations of them.
However, similar thoughts come again periodically, and a lot of mathematicians would like to have model theory also closer to algebraic geometry. I remember that Angus Macintyre asked in a model theory conference ten years ago "When will we finally dare the step to schemes?"
M^eq is not affine; it is an expansion of the vocabulary to guarantee that quotients are present. For definition of terminology see http://en.wikipedia.org/wiki/Imaginary_element
The examples there of `elimination of imaginary' are dull; more interesting is algebraically closed fields.
The compactness theorem is sometimes used to reduce infinite coverings to finite. See for example Poizat's stable groups for the interaction between definable converings and algebraic geometry.
It depends on the logic you wish to develope. In 1st order logic sometimes it is enough with the stability/categoricity theorems. For sheaf-like logics (where validity is related to open sets), the article of MacIntyre is an excellent introduction. Caicedo's work (in Spanish) is in the same scope, instead of categoricity or stability, here the keystone is the existence of a generic model. An equivariant version of this can be found here. On the other extreme, a categorical approach with topoi and so, is presented in the book of Moerdijk & MacLane.
http://link.springer.com/book/10.1007%2F978-1-4612-0927-0
http://projecteuclid.org/euclid.bsl/1052669289
http://www.accefyn.org.co/revista/Vol_19/74/569-586.pdf
http://arxiv.org/pdf/1304.2477.pdf