$\mu$ is the mobility and $\mathbf{F}_i$ is the inter-particle repulsion force accounting for excluded volume between particles which derives from a continuous potential $$\mathbf{F}_i= - \nabla_i \sum_{j < i} v(r_{ij}=|\mathbf{r}_i-\mathbf{r}_j|),\, v(r)= \epsilon \left(\frac{\sigma}{r}\right)^{12}\, ,$$ with an upper cutoff at $3\sigma$. The parameter $\sigma$ can be thought of as the effective particle diameter. From now on $\sigma=1$, fixing the unit of length. The term $\eta_i$ is a zero-mean white noise with unit-variance. It accounts for fluctuations in the self-propulsion direction, whose strength is controlled by the parameter $\gamma$, introducing time correlations between displacements over a typical timescale $\tau=\gamma^{-1}$, hence called the *persistence time*. The first term in eq. is a continuous-time version of a Vicsek-like interaction: it controls the velocity alignment between the particles, leading to local orientational ordering with a coupling strength $K$ (normalized by the interaction area, see below). This coupling strength defines the characteristic time-scale associated to the phase dynamics $\tau_K=\pi R_{\theta}^2/K$. Here, in the spirit of previous modeling of ABP with alignment interactions [@Peruani2008mean; @Chepizhko2010; @Farrell2012; @CAP; @CAPhard], we introduce a ferromagnetic-like coupling between velocities akin to the one of the Kuramoto model of phase synchronization [@KuramotoRev] or, equivalently, the XY model in two dimensions. The sum 