  We discuss the classical problem of counting planes tangent to general
canonical sextic curves at three points. We determine the number of real
tritangents when such a curve is real. We then revisit a curve constructed by
Emch with the greatest known number of real tritangents, and conversely
construct a curve with very few real tritangents. Using recent results on the
relation between algebraic and tropical theta characteristics, we show that the
tropicalization of a canonical sextic curve has 15 tritangent planes.
