We present the results of analytical studies and 2D3V PIC simulations of electron-positron plasma cloud collisions. We concentrate on the problem of quasi-static magnetic field generation. It is shown from linear theory, using relativistic two-fluid equations for electron-positron plasmas, that the generation of a quasi-static magnetic field can be associated with the counterstreaming instability. A two-dimensional relativistic particle simulation provides good agreement with the above linear theory and shows that, in the nonlinear stage of the instability, about 5.3% of the initial plasma flow energy can be converted into magnetic field energy. It is also shown from the simulation that the quasi-static magnetic field undergoes a collisionless change of structure, leading to large-scale, long-living structures and the production of high-energy particles. These processes may be important for understanding the production of high-energy particles in the region where two pulsar winds collide.