Can anyone explain the point of this dual PNP/NPN set?
Behavior :

The circuit is a latch.

Assuming the starting voltage across C5 is 0V and Q2 is on, Vbe of Q5 is 0V and will stay that way, unless some other (unshown) part of the circuit pulls up the voltage of the +Vout net, turning on Q5 and then Q4. 

From there, I see two possible answers :


This is a strange way of activating an output from somewhere else, which is overcomplicated;
This is bad design


I (and all the commenters) would favor the second theory, which is confirmed by a few other elements, like the lack of a base resistor on on Q3  or filtering globally.

In Conclusion, check the rest of the schematic for something that could boostrap +Vout, or other examples of copy-pasted circuits that do not belong.