[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/gQsny.jpg

When the Vout is below ground it cannot bias the regulator to start up as the NPN's to ground are reverse biased.

What can you do?
-

Consider the minimum load current 5mA and the idle current 10 mA max and use a pull-up on Vout to Vin+ that forces Gnd to be < Vout+ during V- startup. ( remember Gnd can be floating and just a 0V ref. at some point.) 

How? 
-

- A Zener from Vin to Vout greater than the expected drop which overloads V- until V+ starts? **maybe**  brute force method...

- a CC regulator SMD 20mA chip from Vin to Vout? Not a good idea if the load is less than 20mA then it pulls the Vout higher since it is an emitter follower output.
- Power management cct to ensure both are OK before enabled to IC.  ( **too much trouble**)
- A resistor divider that draws > 10mA  from Vin to Vout and GND so the external voltage V- its pull-up ( more current ) from the V+ being off?   **YES**
- a simple idea may work but depends on dynamic current flow as V- starts up

- compare with a LM317 ? maybe
 
Idle current spec : *Vin ≤ 8V   5(typ) 10 mA (max)*  

Remember LDO's only source current and not sink.   
But if the output is reverse biased relative to GND **they cant source either** because the Gnd is a controlled current **sink**.

Best bet
-

- get rid of the V- supply altogether and use  Charge pump IC to generate V- from V+ only.