  In this paper we introduce the notion of $a$-monoidal distributive law
between two Hopf quasigroups $A$ and $H$. We prove that every $a$-monoidal
distributive law induce a product on $A\otimes H$, called the wreath product,
thanks to which $A\otimes H$ becomes in a Hopf quasigroup. Finally, using this
construction, we show that double cross products of Hopf quasigroups, cross
products of Hopf quasigroups with a skew pairing between them, Hopf quasigroups
defined by the twisted double method, smash products of Hopf quasigroups and
twisted smash products of Hopf quasigroups are examples of wreath products
associated to $a$-monoidal distributive laws.
