1967 Plymouth Barracuda - Fuel tank not vented properly??????

Will retrain myself to not overfill the fuel tank on the Cuda!
Pretty much all cars. Worst is winter fuel on a hot day. Look up "winter fuel RVP". Although what we're seeing when liquid fuel forces out is probably not the RVP directly, but regardless it expands.

no check valve.

You seem to have a vent system that is older than your car. up through 1970 the vent was simple and ended inside the rear frame rail. it is actually design not to vent anything out of the tank, but rather let air into the tank to balance the fuel being pumped out.
^^This^^

The vent line goes back into the fuel fill. The flip cap itself should allow some air into the tank. Not so on later models which vent through intermediate devices - 1975 is probably a charcoal filter.

Later cars also seal up the air cleaner snorkel when the engine is shut off and vent the carb bowl to the charcoal canister when the engine is off. That might after '75.