In my answer at What, in simplest terms, is gauge invariance?, I mentioned that in certain contexts there can be a "gauge theory" with a local symmetry that leave the Lagrangian/Hamiltonian invariant and whose degrees of freedom are physically observable. My follow-up got very long, so I decided to address the Lagrangian and Hamiltonian cases separately. Here are my thoughts on the Lagrangian case - anyone else is of course welcome to contribute their own. The case of lattice gauge Hamiltonian is addressed at Hamiltonian lattice gauge theory with physically observable local degrees of freedom.

In my answer at What, in simplest terms, is gauge invariance?, I mentioned that in certain contexts there can be a "gauge theory" with a local symmetry that leave the Lagrangian/Hamiltonian invariant and whose degrees of freedom are physically observable. My follow-up got very long, so I decided to address the Lagrangian and Hamiltonian cases separately. Here are my thoughts on the Lagrangian case - anyone else is of course welcome to contribute their own. The case of lattice gauge Hamiltonian is addressed at Hamiltonian lattice gauge theory with physically observable local degrees of freedom.

In my answer at What, in simplest terms, is gauge invariance?, I mentioned that in certain contexts there can be a "gauge theory" with a local symmetry that leave the Lagrangian/Hamiltonian invariant and whose degrees of freedom are physically observable. My follow-up got very long, so I decided to address the Lagrangian and Hamiltonian cases separately. Here are my thoughts on the Lagrangian case - anyone else is of course welcome to contribute their own thoughts.

In my answer at What, in simplest terms, is gauge invariance?, I mentioned that in certain contexts there can be a "gauge theory" with a local symmetry that leave the Lagrangian/Hamiltonian invariant and whose degrees of freedom are physically observable. My follow-up got very long, so I decided to address the Lagrangian and Hamiltonian cases separately. Here are my thoughts on the Lagrangian case - anyone else is of course welcome to contribute their own. The case of lattice gauge Hamiltonian is addressed at Hamiltonian lattice gauge theory with physically observable local degrees of freedom.

A gauge theory is usually defined as containing degrees of freedom that are both local and physically unobservable, but some physicists use a more general definition that only requires the degrees of freedom to be physically unobservable, not necessarily local. Are there any "gauge theories" which make the opposite generalization - i.e. they have local degrees of freedom that leave the Lagrangian/Hamiltonian unchanged, but which are physically observable?

(This post is a follow-up to my answer at What, in simplest terms, is gauge invariance?. I will post my own answer below, although of course anyone who has any other thoughts is free to answer as well.)

In my answer at What, in simplest terms, is gauge invariance?, I mentioned that in certain contexts there can be a "gauge theory" with a local symmetry that leave the Lagrangian/Hamiltonian invariant and whose degrees of freedom are physically observable. My follow-up got very long, so I decided to address the Lagrangian and Hamiltonian cases separately. Here are my thoughts on the Lagrangian case - anyone else is of course welcome to contribute their own thoughts.