In this paper, assuming multi-antenna transmitter and receivers, we consider multicast beamformer design for the weighted max-min-fairness (WMMF) problem in a multi-stream multi-group communication setup. Unlike the single-stream scenario, the WMMF objective in this setup is not equivalent to maximizing the minimum weighted SINR due to the summation over the rates of multiple streams. Therefore, the non-convex problem at hand is first approximated with a convex one and then solved using Karush-Kuhn-Tucker (KKT) conditions. Then, a practically appealing closed-form solution is derived, as a function of dual variables, for both transmit and receive beamformers. Finally, we use an iterative solution based on the sub-gradient method to solve for the mutually coupled and interdependent dual variables. The proposed solution does not rely on generic solvers and does not require any bisection loop for finding the achievable rate of various streams. As a result, it significantly outperforms the state-of-art in terms of computational cost and convergence speed.