When is a 0-1 matrix a one-intersection incidence matrix?

The following problem is what motivated my previous MO question.

It is easily seen that for any given 0-1 matrix $M$, one can always find a set $\mathcal P$ of points, and a set $\mathcal C$ of simple curves in the plane, so that their incidence matrix is exactly the matrix $M$. Suppose, however, that any pair of curves from $\mathcal C$ is now allowed to intersect in at most one point (be it a point of $\mathcal P$ or any other point), and let's say that the matrix $M$ is realizable if such $\mathcal P$ and $\mathcal C$ can be found. Clearly, a necessary condition for this is that the scalar product of any two rows of $M$ be at most $1$, but this condition is insufficient: say, for $q$ large enough, by the Szemeredi-Trotter theorem, the point-line incidence matrix of the finite projective plane $PG(2,q)$ has two many incidences to be realizable. What are other reasonable necessary / sufficient conditions for $M$ to be realizable? What are "small" examples of non-realizable 0-1 matrices?

Here is a very specific question along these lines. Everything I presently can say on the problem above is essentially symmetric in $\mathcal P$ and $\mathcal C$. This suggests that, perhaps, there is some duality between points and curves involved. Accordingly, I wonder whether, by any chance, it can be true that $M$ is realizable if and only if its transpose $M^t$ is realizable?