[en] The ${\mathbb{C}\mathrm{P}}^{N-1}$ model in 2D is an interesting toy model for 4D QCD as it possesses confinement, asymptotic freedom and a non-trivial vacuum structure. Due to the lower dimensionality and the absence of fermions, the computational cost for simulating 2D ${\mathbb{C}\mathrm{P}}^{N-1}$ on the lattice is much lower than that for simulating 4D QCD. However, to our knowledge, no efficient algorithm for simulating the lattice ${\mathbb{C}\mathrm{P}}^{N-1}$ model for $N>2$ has been tested so far, which also works at finite density. To this end we propose a new type of worm algorithm which is appropriate to simulate the lattice ${\mathbb{C}\mathrm{P}}^{N-1}$ model in a dual, flux-variables based representation, in which the introduction of a chemical potential does not give rise to any complications. In addition to the usual worm moves where a defect is just moved from one lattice site to the next, our algorithm additionally allows for worm-type moves in the internal variable space of single links, which accelerates the Monte Carlo evolution. We use our algorithm to compare the two popular ${\mathbb{C}\mathrm{P}}^{N-1}$ lattice actions and exhibit marked differences in their approach to the continuum limit.