I want to generate random rectangular partition of a given $m*n$ rectangle under the constraint that it must be nowhere-neat partition. Nowhere-neat partition means that a dissection of a rectangle into smaller rectangles such that the original rectangle is not divided into two subrectangles.
My interest is in generating rectangular partitions that satisfy these two conditions: 1- No two adjacent rectangles share a common side (i.e. the union of two adjacent rectangles is not a rectangle). 2- At most two corners meet at a point.
Similar partitions appear under other names that include rectangulations, Tatami tilings, and non-slicing floorplans.
What algorithms are there that generate random fault-free rectangular partitions?
Update: Here is an example of random Mondrian rectangular partitions. Here the original rectangle is divided into two subrectangles. However, it is the opposite of what I want.
I do not require uniform distribution. I do not require a specific number of tiles, I only require that the number of tiles $T$ is within some fixed ratio $\epsilon \lt 1$ of $mn$ where $m= \theta(n)$ ($T=\epsilon*m*n$).