Is there a 4 open emitter/4 open collector Darlington pair chip?
Surprisingly, functionally, yes, there is, but it's probably not what you want  - see at end.

ST octal high/low side driver
 2 x low side
 6 x high or low side.

They say 


The L9848 IC is a highly flexible monolithic 
medium current output driver that incorporates 2 
dedicated low side outputs (outputs 7-8) and 6 
outputs that can be used as either internal low or 
high side drives in any combination (outputs 1-6).
In addition, 2 outputs are capable of being 
PWMed via an external pin (outputs 5-6). The 
integrated standard serial peripheral interface 
(SPI) controls all outputs and provides diagnostic 
information.
Integrated clamping circuits, waveshaping, 
protection against positive and negative voltage 
transients and thermal shutdown for all outputs 
open a wide range of automotive and industrial 
applications




BUT this may not suit you - It's SPI controlled, voltage or current may be inadequate, or ... ?

What are you trying to achieve?
 What voltages.
 What currents.
 The high side river you specified had low side inputs built in - this is very nice but is far from implicit in your description. Is that part of your spec?

You need to provide  more detail of what you want to do, with less emphasis on how you think it should be done. eg you say "Darlington". Why?  The L9848 uses MOSFETs - Darlington unneeded as current gain not an issue. The 6 hi/lo FETs are floating so can be effectively open drain or source. The drive is low side regardless - this may suit you very well OR may not. You need to say.