Tar paper and insulation?

[Lingwendil]

Hey all. I have a approximate 12x16 mother in law unit/ workshop that is fully framed, bare studs, and has an attic area rafters up, and I'm in the process of modifying a few things, as well as prepping for insulation. Plan is insulation, drywall, and to set it up as a home office and game room.

While running electrical and pulling nails I'm finding that the exterior is plywood with a layer of black *tar* type paper between the studs and plywood, with mesh and stucco on the exterior of the building.

My question here, is when choosing fiberglass insulation batts for the walls, would I want to leave this black paper intact or remove it? And then, would I want to use unfaced or faced batts? I'm planning on R13, and have 100 square feet roll of faced R13 fiberglass insulation on hand already if that matters

For reference, I live in the east San Francisco Bay area (Contra Costa county) Antioch, California. Highs of 110 degrees in summer, 35-40degrees in winter. I plan to use a small AC unit mounted in the wall in summer, and a small space heater in winter if needed.

Some pics of the space, in progress of cleaning up so pardon the mess.

 

[WisJim]

Before building wraps like Tyvek, tarpaper could do the job of an air barrier if well installed. I'd leave it if it is in good condition.

 

[Red 17]

Leave it, it is also the moisture barrier as water will find it's way through stucco.

 

[Lingwendil]

Would I still want to use batts with kraft facing before I install drywall?