Simulation using Phillips triangular representation and Monte Carlo simulation paths are very different things. The former uses additional linearity in a binary base class (notice $\mathbb{I(0)}$ to generate $\mathbb{I(1)}$) to derive the constitutional elements of the $\mathscr{spread_t}$ and the latter relies very heavily on (preferably) multidimensional random walks, in a reverse-way of the usual statistics. I have also found this new method for simulation .

I've searched the pdf for Monte Carlo but I couldn't find any. This is 50 pages long. Can you pinpoint the section or page number where the possibility of simulation has been hinted or given?

I have added a note in the answer. Hope this helps! Cheers!