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The formation of thermoacoustic shocks is studied in a fluid complex plasma. The thermoacoustic wave mode can be damped (or anti-damped) when the contribution from the thermoacoustic interaction is lower (or higher) than that due to the particle collision and/or the kinematic viscosity. In the nonlinear regime, the thermoacoustic wave, propagating with the acoustic speed, can evolve into small amplitude shocks whose dynamics are governed by the Bateman-Burgers equation with an additional nonlinear term that appears due to the particle collision and nonreciprocal interactions of charged particles providing the thermal feedback. The appearance of such nonlinearity can cause the shock fronts to be stable (or unstable) depending on the collision frequency remains below (or above) a critical value and the thermal feedback is positive. The existence of different kinds of shocks and their characteristics are analyzed analytically and numerically with the system parameters that characterize the thermal feedback, thermal diffusion, heat capacity per fluid particle, the particle collision and the fluid viscosity. A good agreement between analytical and numerical results is also noticed.