I've been unable to get over the hurdle of understanding this concept. However, I'm willing to bet that once I finally understand it will all seem obvious.
I do understand that the real part of the curve divides the complexification of the curve into two halves.
What I've been unable to observe is that the two complex halves have "opposite orientations", but I am willing to bet it will become obvious once I understand what is meant by the "canonical orientation" of each of the two complex halves.
In particular, I would greatly appreciate a simple example of a curve which makes the orientations easy to observe.
The following is from page 8 of this document: http://www.pdmi.ras.ru/~olegviro/introMSRI.pdf
"If the real part of a curve divides its complexification, then each of the halves has the canonical orientation defined by the complex structure and defines an orientation on the common boundary of the halves, which is the set of real points. Thus the set of real points receives two orientations. It is easy to see that these orientations are opposite to each other. They are called the complex orientations of the curve"
"A collection of $n$ ovals can be oriented in $2^n$ distinct ways. Therefore, if the number of ovals is greater than one, the pair of complex orientations is a new structure on the real part of the curve which comes from the complex domain. If one insists on the purely real viewpoint, ignoring all complex phenomena, the complex orientations look mysterious: a choice of orientation of one of the ovals determines the orientations of all the other ovals"
I do have a guess as to what is meant by the "canonical orientation" of each of the two complex halves
The complex plane carries a natural orientation, where multiplication by $i$ rotates counterclockwise.
My guess is that one of the complex halves can be considered to be fundamentally different from the other half; while multiplying by $i$ corresponds to counterclockwise rotation in one complex half, multiplying by $i$ corresponds to clockwise rotation in the other complex half
However, I still have trouble visualizing what this would look like and how/why it would be the case.
Do the two halves have opposite chirality? If so, in what sense? And why?