  We study the behavior of theta characteristics on an algebraic curve under
the specialization map to a tropical curve. We show that each effective theta
characteristic on the tropical curve is the specialization of $2^{g-1}$ even
theta characteristics and $2^{g-1}$ odd theta characteristics. We then study
the relationship between unramified double covers of a tropical curve and its
theta characteristics, and use this to define the tropical Prym variety.
