What is a term for the long-term emotional "climate" of a relationship?

Based on the description and example sentence, a good word is nature:

[Merriam-Webster]
1 a : the inherent character or basic constitution (see CONSTITUTION sense 2) of a person or thing : ESSENCE
       // the nature of the controversy
1 b : DISPOSITION, TEMPERAMENT
       // it was his nature to look after others
       — F. A. Swinnerton
       // her romantic nature
3 : a kind or class usually distinguished by fundamental or essential characteristics
    // documents of a confidential nature
    // acts of a ceremonial nature

Used in the example sentence in the question:

At that moment, the mood between them was tense, even though in general the nature of the relationship was carefree.

The word "nature" comes to mind.

According to Webster's New Dictionary of Synonyms p. 839:

nature may suggest inherent, essential characteristics rather than superficial, ostensible, or tentative ones

This means that you can use "nature" to describe the "climate" but "mood" for the "weather."

An emerging term/slang is using the word meta as a noun to represent the general popular theme/ideas of something, typically a video game.

For example,

In 2014, the Minecraft meta was all about building castles, expansive minecart tracks, and cactus farms.

So you could potentially refer to it as the meta of their relationship.

The meta of their relationship was carefree.

You could also consider tenor

Defined by Cambridge as:

the general meaning, character, or pattern of something: What was the general tenor of his speech?

In your sentence:

At that moment, the mood between them was tense, even though in general the tenor of the relationship was carefree.

The two JBs have suggested nature, which is indeed an excellent word.

However I would suggest the relationship itself is carefree - you might say the relationship is in general characterised by being carefree, or simply though their relationship was generally carefree.