We study a generalization of the asymmetric simple inclusion process (ASIP) on a periodic one-dimensional lattice, where the integers in the particles rates are deformed to their t-analogues. We call this the \((q, t, \theta )\) ASIP, where q is the asymmetric hopping parameter and \(\theta \) is the diffusion parameter. We show that this process is a misanthrope process, and consequently the steady state is independent of q. We compute the steady state, the one-point correlation and the current in the steady state. In particular, we show that the single-site occupation probabilities follow a beta-binomial distribution at \(t=1\) . We compute the two-dimensional phase diagram in various regimes of the parameters \((t, \theta )\) and perform simulations to justify the results. We also show that a modified form of the steady state weights at \(t \ne 1\) satisfy curious palindromic and antipalindromic symmetries. Lastly, we define an enriched process at \(t=1\) and \(\theta \) an integer which projects onto the \((q, 1, \theta )\) ASIP and whose steady state is uniform, which may be of independent interest.
