What is the gender-neutral way of saying “gentlemen’s agreement”?

A verbal agreement is a phrase you could use; an informal agreement would be another.

Wikipedia defines it as follows:

A gentlemen's agreement, or gentleman's agreement, is an informal and legally non-binding agreement between two or more parties. It is typically oral, but it may be written or simply understood as part of an unspoken agreement by convention or through mutually-beneficial etiquette.

Its main characteristics are that it’s not legally enforceable, and thus based on trust.

https://en.m.wikipedia.org/wiki/Gentlemen's_agreement provides details on the use of the term in commerce and trade.