Simeon Hellerman from the IAS at Princeton spoke about the landscape of nongeometric compactifications and it was very interesting.
At the beginning, there was some debate what class of vacua he really wants to consider and what belongs to this class and what does not. After a short time, it became clear what theories he actually wants to talk about: perturbative, locally geometric vacua.
The extra dimensions are not really compactified on manifolds but just on "things". (I made a joke and asked whether these "things" are "categories", and the answer was essentially "Yes".) What is Simeon's "thing"? It is a fibration over a base space in which the fiber can undergo not only SL(k,Z) monodromies around codimension 2 singular loci of the base space, but more general SO(k,kZ) T-duality-like transformations. His condition is, however, that all these monodromies around the "minimal" singularities in the base space should be conjugate to elements of SL(k,Z), so that the local physics in the base space is equivalent to geometric physics via dualities.
In the case of the three-tori, k=3, the picture above is an ideology. In the case of k=2, the allowed conjugations must include the exchange of the complex structure "tau" and the complexified Kähler structure "rho" if the conjugations are supposed to generate non-trivial backgrounds. Simeon can define very a nice thing, the so-called Thing-1 or \Theta1 for short. More precisely, he can generalize the concept of an eliptically fibered K3 manifold in an interesting way. Such a non-manifold has 24 singular fibers and the monodromy around each of them acts on the complex structure "tau" or the complexified Kähler form "rho" nontrivially.
It turns out that there are two more choices beyond the ordinary K3 manifold in which only "tau" is affected. One can have 24 singular points around which there is a monodromy induced on "rho", the complexified Kähler form of the torus: if you go around a point, you induce a T-duality transformation. The resulting space is a semi-mirror of a K3 and there is no new physics.
However, you can also have a manifold with 12 singular fibers with a "tau" monodromy and 12 singular fibers with a "rho" monodromy. You can think of this structure as a conglomerate of two glued half-K3's such that one of which has been half-mirrored before you glued them. One can derive the spectrum of massless fields in this structure used for 4 dimensions of type II string theory. It turns out that you preserve one quarter of supersymmetry (instead of one half, as in the case of a K3) and one obtains exotic numbers of hypermultiplets, vector multiplets, and tensor multiplets (plus a gravity multiplet, of course). The fact that the 6D anomaly cancellation condition with 273 on the right hand side is satisfied is good enough evidence for me that it is probably a consistent background.
Simeon can then analyze the spectrum of D-branes and the corresponding RR charges. This spectrum somewhat resembles what you would expect from a heterotic string: for example, only the right-moving sector of the string contains spacetime supercharges which means that the superpartners of the NS-NS states only appear in the NS-R sector but not the R-NS sector.
Finally, Simeon has answered a couple of questions about the number of such exotic vacua. He thinks that they are generic, they seem more general and have the ability to contradict some of Cumrun-like general predictions of the landscape (as opposed to the swampland) - which Cumrun denies because there is no known example that actually violates them - and Simeon also confirms my comment that with this new notion of genericity, it is more reasonable to say that the typical vacua tend to have the fibers comparable to the string scale which would disfavor "old large dimensions" and similar scenarios.