To improve the dietary habits of the population, the EU, within the Farm to Fork strategy (F2F), is strongly supporting the Nutri-Score (NS) Front Of Pack (FOP) label. Under the NS system, Geographical Indications (GIs) are generally scored as “unhealthy” food, given the predominance of products of animal origin among GIs which are, notoriously, high-fat products. This study aims to determine the impact of the NS label on consumers’ preferences for two Protected Designation of Origin (PDO) cheeses, in comparison with generic ones. A Discrete Choice Experiment (DCE) was conducted on 600 Italian consumers through the estimation of a Random Parameter Logit model. Results highlighted that Italian consumers are generally not familiar with the NS and perceive it as a positive characteristic of the product, even if it is signalling an unhealthy choice (D score). However, consumers aware of the Nutri-Score meaning are willing to pay less to buy a product considered “unhealthy” according to this system. Furthermore, we found that consumers who already knew the NS system have homogeneous behaviours in rejecting the product, independently of the association with a PDO certification. This result has important implications on the agri-food sector. If the Nutri-Score becomes mandatory in the EU, consumers might refuse many GIs due to their negative Nutri-Score values. However, the quality of these products is recognized and protected worldwide. In this vein, the GI policy could be questioned by the F2F strategy: both of them aims to reduce information asymmetry producing, at the same time, contrasting results. Within the Geographical Indication policy, the PDO and PGI goods are protected for their quality attributes, which are strictly linked to their geographical origin of the products and traditional know-how. However, the EU adoption of the Nutri-Score could damage these products, reducing their perceived quality/value.