Geometrically this corresponds to a variety with only one analytic branch at a point.
There are many examples of reduced and irreducible local rings that are analytically reducible, such as the local ring of a node of an irreducible curve, but it is hard to find examples that are also normal.
Nagata (1958, 1962, Appendix A1, example 7) gave such an example of a normal Noetherian local ring that is analytically reducible.
Suppose that K is a field of characteristic not 2, and K [[x,y]] is the formal power series ring over K in 2 variables.
Let R be the subring of K [[x,y]] generated by x, y, and the elements zn and localized at these elements, where Then R[X]/(X 2–z1) is a normal Noetherian local ring that is analytically reducible.