  We expand our previously founded basic theory of equiresidual algebraic
geometry over an arbitrary commutative field, to a well-behaved theory of
(equiresidual) algebraic varieties over a commutative field, thanks to the
generalisation of affine algebraic geometry by the use of canonical
localisations and $*$-algebras. We work here in an equivalent and more
suggestive "concrete" setting with structural sheaves of functions into the
base field, which allows us to give a set-theoretic description of the products
of equivarieties in general. Locally closed subvarieties are naturally equipped
with an equivariety structure as in the particular case of algebraically closed
fields, and this allows in particular to embed all quasi-projective
(equi)varieties in general into the category of algebraic (equi)varieties.
