  In the Euclidean setting, the Fujii-Wilson-type $A_\infty$ weights satisfy a
Reverse H\"older Inequality (RHI) but in spaces of homogeneous type the best
known result has been that $A_\infty$ weights satisfy only a weak Reverse
H\"older Inequality. In this paper, we compliment the results of Hyt\"onen,
P\'erez and Rela and show that there exist both $A_\infty$ weights that do not
satisfy an RHI and a genuinely weaker weight class that still satisfies a weak
RHI. We also show that all the weights that satisfy a weak RHI have a
self-improving property but the self-improving property of the strong Reverse
H\"older weights fails in a general space of homogeneous type. We prove most of
these purely non-dyadic results using convenient dyadic systems and techniques.
