My sub keeps breaking protocol when we're around other people and then apologising privately afterward. I don't know if this is accidental or a kind of brat behaviour. How do I address it?

The pattern you are describing, consistent failure followed by private apology, has a few possible explanations, and which one is operating matters for how you address it.

One possibility is that your protocols are easier to maintain in private than in public because social context activates different automatic behaviours. Many submissives find that switching between public-facing social self and dynamic-compliant self is genuinely difficult, particularly early in a dynamic or in social situations that are ambiguous or stressful. This is not defiance; it is a real calibration challenge.

Another possibility is that this is a form of seeking a response from you. The apology makes the pattern visible, which means your sub is aware of it and has chosen to bring it to your attention repeatedly without the dynamic producing the response that would actually address it. That can be a form of seeking engagement, attention, or consequences within the dynamic.

A direct conversation outside any scene context, asking your sub to describe in their own words what happens when they break protocol in public and why they think it is happening, gives you actual information. Most people can tell you something useful about their own pattern if asked directly and non-punitively.

From that conversation you can decide whether the protocol needs to be adjusted for public contexts, whether there is a specific consequence structure that would help, or whether there is something about the dynamic in public that is not working that needs rethinking.

Addressing it through a clear plan is more effective than addressing each individual breach after the fact.