  We briefly review and critically compare three approaches to constructing
Lagrangian theories of self-interacting Abelian chiral form fields with
manifest Lorentz invariance. The first approach relies on the original ideas of
Pasti, Sorokin, and Tonin (PST) and has been explored since the late 1990s. The
second approach was introduced by Ashoke Sen in 2015. The third approach has
been developed over the last few years in the works of the present authors and
other collaborators and may be called the 'clone field' formalism since it
features an auxiliary 'clone' of the gauge field sector. We argue that this
last approach shares the attractive features of the other two while avoiding
their respective shortcomings. Like in Sen's approach, within the clone field
formalism, arbitrary interactions can be straightforwardly included in any
number of dimensions (treating interactions becomes very difficult in the PST
formalism in dimensions greater than 6). Like in the PST approach, all the
auxiliary fields are gauged away on-shell (while in Sen's approach, they merely
decouple from the physical fields but remain dynamical).
