Why this saw shaper circuit does not work in reality?
The input waveforms you're applying in your real circuit do not match those you're applying in the simulation.
In the sim, the square wave edges are aligned to the peaks of the triangle wave, but in your real circuit the screenshot shows that the square wave edges are aligned to the zero-cross points of the triangle wave.
This results in the opamp's configuration being switched at the wrong points in the triangle.