Quick rundown of my process...
I collect 20 gallons of RO water in my HLT.
I add all of the minerals for mash and sparge water to that volume.
I heat that water to roughly 6 degrees over my strike temp.
Transfer my desired mash volume to MLT.
I mix in my desired amount of baking soda/pickling lime with the grains before dough in.
Once I add the grains to the MLT, I add whatever amount of cool water necessary to get my HLT back up to 17 gallons (this is where the HERMS coil is completely covered).
I then add back more minerals to the sparge water based on how much water I added back in. So if I added 6 gallons of water to the HLT, I'll take the per gallon numbers from Bru'n Water, do the math and add that to HLT.
In answer to your question, yes, I fly sparge. On this particular batch, took me approx. 90-95 minutes to complete the sparge.
I have not checked the pH of the sparge runnings, only the mash. Mash pH here was 5.45.
I can say that during the mash, the wort did not taste astringent.
During the boil, it did not taste astringent. I can say that the first imperial stout I did, I added baking soda to the sparge water by mistake. I can verify that you CAN taste astringency during the boil.
However, in this case, I didn't pick it up until after fermentation, which I thought was weird.