We propose a Monte Carlo code for the simulation of the fragmentation process of polarized quarks into pseudoscalar mesons. Such process is generated recursively once the flavour, the energy and the spin density matrix of the initial quark are specified, performing a cascade of splittings of the type $q\rightarrow h+q'$, where $q, q'$ indicate quarks and $h$ the hadron with flavour content $q\bar{q}'$. Each splitting is generated using a splitting distribution which has been calculated in a string fragmentation framework including, for the first time, the ${}^3P_0$ mechanism. This mechanism involves a "complex mass" parameter responsible for transverse spin effects, such as the Collins effect. Results for single hadron and hadron pair analyzing power are found to be in agreement with experimental results from SIDIS and $e^+e^-$ annihilation.