To the best of my knowledge, the ASI is not calibrated to take compressibility into account. This is the reason we have EAS. in the first place.
ASIs have been designed to take account of compressibility since the 1930s. Equivalent airspeed has absolutely nothing to do with compressibility. EAS is exactly as QLB has pointed out: a value, expressed in units of speed, associated to a dynamic pressure. The lynchpin in understanding why it has nothing to do with compressibility lies in acknowledging that dynamic pressure assumes the fluid is incompressible.
If the pressure in the pitot tube was incompressible, TAS and EAS would always be equal.
EAS = TAS * sqrt(rho/rho_0), contains the variable rho (air density) whose variation with altitude leads to a divergence between TAS and EAS. (rho_0 is ISA MSL rho).
There are plenty of rigorous treatments of this topic (John D Anderson, Liepmann & Roshko, etc). W. Aiken's 1946 NACA Report 837 is freely available at Cranfield's NACA mirror at this URL:
Standard nomenclature for airspeeds with tables and charts for use in calculation of airspeed