Probably too late to help your particular issue, but here goes.
First, there are a lot of possible ternary phase diagrams, most of which are not technologically useful (yet), so there isn't much funding to go look at them. Heck, there are many binary phase diagrams that are not really well researched (I was surprised this week to find that the Pd-Pt diagram doesn't have much work). Phase diagrams look easy, but the thermodynamic measurements behind them are difficult, and you need a good reason.
That being said, what is available?
On the ASM International Alloy Phase Diagram Database (subscription required, may be available through your institution), there is not entry for Au-Bi-Ga. That means no experimental or calculated analyses are indicated there. This is not definitive, since I have found many articles in the literature that are not in their database. But, it is generally a good place to start. Better searches for Au-Bi-Ga return nothing. So, we need to look to the binary literature and see what might be what.
For the binary diagrams, data exist for all of them: Au-Bi, Au-Ga, and Bi-Ga, with thermodynamic assessments of the CALPHAD type.
Au-Bi - from C. Servant et al, Calphad 30(4), 443-448 (2006). There is one intermetallic phase, Au2Bi.
Au-Ga - from J. Wang et al., Calphad 35, 242-248 (2011). There are 6 intermetallic phases.
Bi-Ga - from F. Sommer, Z. Metallkd. 87, 865-873 (1996). No intermetallics, but a liquid miscibility gap.
All three elements have different elemental crystal structures (fcc, hcp, ortho for Au, Bi, and Ga). None of the Au-Ga intermetallics have a crystal structure similar to Au2Bi.
A ternary phase, dissimilar to any of the intermetallics, while possible would seem unlikely. Extensive solubility of one element into the 'opposite' binary intermetallic (e.g. Ga into Au2Bi) also seems unlikely given the different base crystal structures. In other words, while I would expect Ag solubility into Au2Bi (mixing with the Au sub-lattice) because it is also an fcc metal, the very limited solubilities on the binary diagrams indicates there is little solubility into the respective intermetallics.
So, no, there are no available analyses of the ternary system. Perhaps in the course of your project you can add to the thermodynamic knowledge. Good luck with your work.