  We show that totally real elliptic Lefschetz fibrations that admit a real
section are classified by their "real loci" which is nothing but an
$S^1$-valued Morse function on the real part of the total space. We assign to
each such real locus a certain combinatorial object that we call a
\emph{necklace diagram}. On the one hand, each necklace diagram corresponds to
an isomorphism class of a totally real elliptic Lefschetz fibration that admits
a real section, and on the other hand, it refers to a decomposition of the
identity into a product of certain matrices in $PSL(2,\Z)$. Using an algorithm
to find such decompositions, we obtain an explicit list of necklace diagrams
associated with certain classes of totally real elliptic Lefschetz fibrations.
Moreover, we introduce refinements of necklace diagrams and show that refined
necklace diagrams determine uniquely the isomorphism classes of the totally
real elliptic Lefschetz fibrations which may not have a real section. By means
of necklace diagrams we observe some interesting phenomena underlying special
feature of real fibrations.
