Equivalent english of Brazilian expression about trusting someone "by the mustache"?

In Brazilian Portuguese, we use the following expression:

Fio do bigode (mustache wire/thread or mustache hair)

Here's what I found after researching:

It consisted of a man giving as a warranty, for his word, a thread of his beard, usually removed from his mustache. Of controversial origin, mustache may have come from the old German expression pronounced in the oaths: "bi Gott", or "by God"

Example:

O contrato entre as duas partes foi firmado pelo fio do bigode.

Which Google Translate readily translates to:

The contract between the two parties was signed by the mustache.

The above expression means that both parties agreed to something without actually signing a written contract or equivalent. Basically, both parties trusted each other by the other's history/legacy/references/appearance/influence/promises or anything that might have transmitted confidence.

What would be an equivalent English expression?

A possible expression is a gentleman's agreement:

[Merriam-Webster]
variants: or gentlemen's agreement
: an agreement secured only by the honor of the participants
// That also ties into recent speculation that Chelsea have a gentleman's agreement not to pursue Napoli players.
— SI.com, "Chelsea Submit 'Important Offer' for Juventus Defender as Summer Business Finally Gets Going," 22 June 2018

These are sometimes legally enforceable, and sometimes not. It depends on the context.

In the age of knighthood and chivalry (at least according to its reputation in media and popular culture), honour was a much more significant commodity than it seems to be today. Assuming that was actually the case, then saying on my honour, especially where witnesses were involved, would have been almost as good as actually signing something.

In the English-speaking world, we don't typically make deals by exchanging mustache hairs, but we do make deals with handshakes.

An equivalent expression could be a handshake deal/agreement.

A handshake agreement is an agreement to do something based only on good faith, which cannot be legally enforced. Two partners might make a handshake agreement in which they both promise to work hard on a joint project. These types of agreements can significantly improve behavior in some settings.

Knowledge @ Warton High School

A handshake deal is a verbal commitment to a transaction. The actual transaction comes later, when documents are signed and money changes hands.

Y Combinator

A more formal term of roughly the same meaning would be an oral contract.

There are several phrases in English that might cover this:

A handshake deal or oral contract is one that is agreed to without a written instrument; under certain circumstances, U.S. courts have held them to be as binding as written contracts.

To say that one’s word is one’s bond (definition from Cambridge) (definition from Merriam-Webster) is a somewhat dated expression that says that a person will always keep a promise once made.