  We give the addition formula for the tropical Hesse pencil, which is the
tropicalization of the Hesse pencil parametrized by the level-three theta
functions, via those for the ultradiscrete theta functions. The ultradiscrete
theta functions are reduced from the level-three theta functions through the
procedure of ultradiscretization by choosing their parameters appropriately.
The parametrization of the level-three theta functions firstly introduced in
\cite{KKNT09} gives an explicit correspondence between the amoeba of the real
part of the Hesse cubic curve and the tropical Hesse curve. Moreover, through
the parametrization, we obtain the subtraction-free forms of the addition
formulae for the level-three theta functions, which lead to the addition
formula for the tropical Hesse pencil in terms of the ultradiscretization.
Using the parametrization, the tropical counterpart of the Hesse configuration
is also given.
