In order to study the pseudo entropy of timelike subregions holographically,
the previous smooth space-like extremal surface was recently generalized to mix
space-like and time-like segments and the area becomes complex value. This
paper finds that, if one tries to use such kind of piecewise smooth extremal
surfaces to compute timelike entanglement entropy holographically, the complex
area is not unique in general. We then generalize the original holographic
proposal of spacelike entanglement entropy to pick up a unique area from all
allowed ``space-like+time-like'' piecewise smooth extremal surfaces for a
timelike subregion. We give some concrete examples to show the correctness of
our proposal.Comment: 26 pages, 10 figure