Safeguarding American Value-added Exports Act of 2023 or the SAVE Act of 2023 This bill directs the Department of Agriculture (USDA) to secure foreign markets for goods using common names. In general, the bill defines common name as a name that USDA determines (1) is ordinarily or customarily used for an agricultural commodity or food product, (2) is typically placed on the packaging and product label of the agricultural commodity or food product, and (3) the use of which is consistent with standards of the Codex Alimentarius Commission. Specifically, USDA must coordinate efforts with the Office of the U.S. Trade Representative (USTR) to secure the right of U.S. agricultural producers, processors, and exporters to use common names for agricultural commodities or food products in foreign markets. Through the negotiation of bilateral, plurilateral, or multilateral agreements, memoranda of understanding, or exchanges of letters, USDA and the USTR must assure the current and future use of each common name identified by USDA in connection with U.S. agricultural commodities or food products.