A plane quartic has 28 bitangents. A tropical plane quartic may have infinitely many bitangents, but there is a natural equivalence relation for which we obtain precisely 7 bitangent classes. If a tropical quartic is Trop(V(q)) for a polynomial q in K[x,y] (where K is the field of complex Puiseux series), it is a natural question where in the 7 bitangent classes the tropicalizations of the 28 bitangents of V(q) are, or, put differently, which member of the tropical bitangent classes lifts to a bitangent of V(q), and with what multiplicity. It is not surprising that each bitangent class has 4 lifts. If q is defined over the reals, V(q) can have 4, 8, 16 or 28 real bitangents. We show that each tropical bitangent class has either 0 or 4 real lifts - that is, either all complex solutions are real, or none. We also discuss further questions concerning tropical tangents, their combinatorics and their real lifts. This talk is based on joint work with Yoav Len, and with Maria Angelica Cueto.