What exactly is an astroglial calcium wave? There are quite a few conflicting reports as to what stimulates them, how they propagate, whether they communicate intercellularly, and what they look like. 

The only consistent information I can find on them is that they propagate through the astrocyte at about 20 microns/sec.

Are they solitons? do they propagate intercellularly? Are they independent of neuronal input? Can the astroglial mesh be viewed as an astroglial field (obviously more speculative)?
I'll try and answer this:

An astroglial calcium wave is the phenomena in which astrocytes express a variety of ion channels and membrane receptors, which enable them to respond on a millisecond time scale to neuronal activity with changes in membrane potential and/or increases in intracellular Ca2+ levels.


  Astrocytes were not only able to respond to external stimulation with increases in intracellular calcium elevations but, most importantly, they were be able to transmit these calcium signals to adjacent non-stimulated astrocytes, as intercellular Ca2+ waves (ICWs). The presence of such phenomenon of propagating waves of calcium lead to the proposition that “networks of astrocytes constitute an extraneuronal pathway for rapid long-distance signal transmission within the CNS.” 


Are they solitons? 

No as they are followed by similar events and a soliton is neither preceded nor followed by another such disturbance.


  A calcium wave is defined as a localized increase in cytosolic Ca2+
  that is followed by a succession of similar events in a wave-like
  fashion.


Do they propagate intercellularly? 

I think so yes or if not propagate then communicate -


  They were be able to transmit these calcium signals to adjacent
  non-stimulated astrocytes.


Are they independent of neuronal input? 


  There is a reciprocal communication between neurons and astrocytes.
  Hippocampal neuronal activity was shown to trigger calcium waves in
  astrocyte networks (Dani et al., 1992) and astrocyte calcium waves
  were shown to modulate neuronal activity (Dani et al., 1992; Kang et
  al., 1998; Nedergaard, 1994; Parpura et al., 1994; Parri et al.,
  2001).


Can the astroglial mesh be viewed as an astroglial field (obviously more speculative)?

This one I couldn't find any discussion on - nor could I form a conclusion on other than I suppose it could?

Reference:

Scemes, E., &amp; Giaume, C. (2006). Astrocyte calcium waves: what they are and what they do. Glia, 54(7), 716-725.