I have recently been thinking about left and right semi-model categories and in which case they can be promoted to Quillen model structure, and I have come to the conclusion that that absolutely all the techniques I have ever seen for constructing full Quillen model structures (by opposition to construction of left or right semi-model structure) relies at some point on an argument that also implies either left or right properness.

A typical example of this is when a model structure is constructed by combining an argument that would usualy produce a left semi-model structure with the fact that every object is cofibrant, but this also implies left properness (that is typically what happen with Cisinki model structures, and their generalization by Olschok).

I'm very curious to know if the I above can be replaced by a we, and to be honest I would be very interested in seeing an example of a construction of model structure that use a very different kind of argument than those I already know. Hence my question:

Is there any known, naturally occurring, example of a Quillen model category that is neither left nor right proper ?

I don't doubt that model structure that are neither left nor right proper exists, in fact I know of one such examples, but it is very artificial and I don't really consider this a good answer to the question: it is the model structure on a four elements poset constructed by Reid Barton as an answer to this recent other question of mine (and that would already be nice), but I'm really more interested in "natural" or "interesting" examples (basically a model structure that could have some interest in its own right, but maybe just an example which is not a poset would already be a first step).