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A \textit{$3$-net} of order $n$ is a finite incidence structure consisting ofpoints and three pairwise disjoint classes of lines, each of size $n$, suchthat every point incident with two lines from distinct classes is incident withexactly one line from each of the three classes. The current interest around$3$-nets (embedded) in a projective plane $PG(2,K)$, defined over a field $K$of characteristic $p$, arose from algebraic geometry. It is not difficult tofind $3$-nets in $PG(2,K)$ as far as $0<p\le n$. However, only a few infinitefamilies of $3$-nets in $PG(2,K)$ are known to exist whenever $p=0$, or $p>n$.Under this condition, the known families are characterized as the only $3$-netsin $PG(2,K)$ which can be coordinatized by a group. In this paper we deal with$3$-nets in $PG(2,K)$ which can be coordinatized by a diassociative loop $G$but not by a group. We prove two structural theorems on $G$. As a corollary, if$G$ is commutative then every non-trivial element of $G$ has the same order,and $G$ has exponent $2$ or $3$. We also discuss the existence problem for such$3$-nets.

3-nets realizing a diassociative loop in a projective plane