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We give an example of a principal algebraic action of the non-commutative free group 
${\mathbb {F}}$
of rank two by automorphisms of a connected compact abelian group for which there is an explicit measurable isomorphism with the full Bernoulli 3-shift action of 
${\mathbb {F}}$
. The isomorphism is defined using homoclinic points, a method that has been used to construct symbolic covers of algebraic actions. To our knowledge, this is the first example of a Bernoulli algebraic action of 
${\mathbb {F}}$
without an obvious independent generator. Our methods can be generalized to a large class of acting groups.
