It may be that my question is answered in Lovász's book:

> Lovász, László. *Graphs and Geometry*. Vol. 65. American Mathematical Soc., 2019.

p.82:
>**Theorem 6.2**. Every planar map in which the unbounded country is a quadrilateral,
all other countries are triangles, and is not separated by a $3$-cycle or a
$4$-cycle, can be represented as a resolved tangency graph of a square tiling of
a rectangle.

Concerning corner touching, Lovász says:

> We can specify arbitrarily one diametrically opposite pair as
'infinitesimally overlapping,' and connect the centers of these two
square[s] but not the other two squares.
We call this a *resolved tangency graph* of the family of squares.

Indeed the source is [Schramm 1993](https://mathoverflow.net/a/378173/6094).