"Under-sign" means to sign or authorize below someone else's signature. However, there are several synonyms for this term. One of them is "counter-sign," which is more commonly used to describe the process of signing a document or check alongside someone else to verify its authenticity. Another related term is "cosign," which refers to the act of signing a loan or credit application together with another person, typically to provide additional guarantee for the lender. Finally, "coauthorize" is another synonym for under-sign, meaning to jointly authorize or approve something with another person or group. Overall, these synonyms all refer to a similar action of signing or authorizing together with someone else to reinforce its validity.