Are base fluid viscosity and grease consistency related?

In general, for most soap thickened greases, the answer is no.  Base fluid viscosity and grease consistency are independent properties of a lubricating grease.  The viscosity of the base fluid is determined by the viscosity of the fluids used, as well as the effect of some additives.  The consistency of a grease is determined by the type and concentration of the thickener in the product.