  We determine the shapes of all degree $4$ number fields that are Galois.
These lie in four infinite families depending on the Galois group and the tame
versus wild ramification of the field. In the $V_4$ case, each family is a
two-dimensional space of orthorhombic lattices and we show that the shapes are
equidistributed, in a regularized sense, in these spaces as the discriminant
goes to infinity (with respect to natural measures). We also show that the
shape is a complete invariant in some natural families of $V_4$-quartic fields.
For $C_4$-quartic fields, each family is a one-dimensional space of tetragonal
lattices and the shapes make up a discrete subset of points in these spaces. We
prove asymptotics for the number of fields with a given shape in this case.
