If you really want to eat where the locals go then the place to visit is the local Sociedad, the equivalent in the UK would be a Community centre or Working Mens club, the only difference is everyone is welcome, no membership, just friendly people serving good food and drink

The local Sociedad to Villa Antonio Lanzarote is on the main road, near enough opposite the mini supermarket in Oasis de Nazaret - worth trying is the "Menu of the Day" 3 courses at a great price

You can find the menu via this link - http://www.villaantonio.co.uk/socidad-menu,

Another place to try in Teguise is La Tahoma, a local bar with food/tapas, the menu is below

Be brave ............ go local!