differential topology - Triangulations of exotic 4-spheres

Here is my comment expanded to answer form: The question of existence of exotic 4-spheres (i.e., the smooth Poincaré conjecture) is still open, and (according to Wikipedia) the existence of exotic PL structures is equivalent to it. Therefore, the answer is that no such explicit triangulations are known.

In general, explicit triangulations of higher dimensional manifolds seem to be difficult to write down. I've heard from computer algebra specialists that no one has even written an explicit triangulation of $mathbb{CP}^3$. The chaos surrounding this earlier question might suggest that the problem is subtle.