MRV is short for measurement, reporting and verification. It describes the entire procedure through which climate projects go to measure the amount of CO2e removed from the atmosphere and report that to an accredited third party. The third party then either verifies or rejects the reporting.
If the results are verified new climate claims can be made. In the case of a Carbon Share a verified reporting would result in an adjustment of the CO2e value per share.
MRV is critical to the quality of a climate project. Without it, it would be impossible to make a trustworthy impact claim. That being said a good MRV system isn't only about high accuracy measurements but also about avoiding potential agency problems by involving an accredited third party.