Based on the expats question "If asked for ID in France by the police, as an EU national what is considered valid?", I know what the rules are for carrying ID as a Brit in France.
Secondly, I know that the Schengen Area (including France and Belgium) doesn't permit systematic checks, and that as an EU National travelling in the Schengen area, I do not need to show an national ID card or passport when travelling from one border-free Schengen EU country to another.
However, as the Europe.eu page says, it's highly recommended to take a passport or ID card with you when travelling between Schengen countries as an EU National.
So, as an EU national, if I start out in north-east France, and cross over into Belgium (on a local train, a bus, a bicycle or similar), what (if any) ID do I need to have with me?