I would imagine a material that is transparent but a electromagnetic wave going out will have a lower frequency than when going in (and maintaining it's direction). You could build glasses to see UV-light with that.
Has an effect like that been observed in nature? Could it theoretically be possible?
Such a thing is possible in principle through the non-linear optical process of half-harmonic generation (and other processes of this class, generally known as photon downconversion).
Usually these non-linear optical processes are rather inefficient. For example, we can talk about the inverse of this process, second-harmonic generation. It is hard to estimate its efficiency because it varies wildly with the input power, but it seems like it may be around 6% for a one-watt infrared pump beam. This is high enough to actually have commercial uses, though. Second-harmonic generation is used to create green lasers. An infrared beam at $1064\,\mathrm{nm}$ is fed through a crystal of KDP, which converts some of the light to $532\,\mathrm{nm}$ green.
In your band of interest of UV, there are some additional considerations (see section "Generating Short Wavelengths") with regards to finding a suitable non-linear medium. But, there definitely are materials that have been used at these frequencies (see here). These examples are all about second-harmonic generation, the opposite of your UV-to-visible - but I think they capture the spirit of the question.
Even if you found a suitable material for downconversion of UV light, there are some issues you'd have in your specific application. One is that the power available in the ambient light field is simply too low. These non-linear effects generally require rather large power densities in order to be noticeable. All of that to say, that although there are indeed materials which have these properties, your specific application idea is infeasible.
Yes, there are various materials that take two or more photons of long wavelength and produce one with a shorter wavelength, and vice versa.
An example of a "step up" material is KTP, which is used in the NIF laser to convert the original infrared laser output to green and then UV. This is more efficient than making a UV laser despite losses in the process.
The opposite direction, "downconversion" is also possible but generally more complex and still under development. But here is one example of doing precisely what you are asking, conversion of UV to visible, in this case green.
