  Consider a Lamperti-Kiu Markov additive process $(J_t,\xi_t:t\geq0)$ on
$\{+,-\}\times\mathbb{R}\cup\infty$ where $J$ is the modulating Markov chain
component. First, we study the finiteness of the exponential functional and
then consider its moments and tail asymptotics under Cramer's condition. In the
strong subexponential case we determine the subexponential tails of the
exponential functional under some further assumptions.
