Oh yea I didn't cover a tie with the RIO player.

In that case it should only be run once because it's the same scenario as if RIO won he just has to split whatever money he won with the other player that tied him.

Edit: nevermind spoke too soon again. Agree it should be playing for a quarter pot as the RIT player that was involved in the side agreed to RIT for his portion of main. So RIO guy gets half of all pots, RIT gets 1/2 of side and 1/4 of main, second board is run for remaining 1/4.

This is correct. For example if the RIO player (player A) has top set, the other big stack (player B) has middle set, and the short stack (player C) has a draw, then player C is actually getting screwed by allowing the main pot to be run twice between B and C (A's equity doesn't change as he only cares about the first runout). If player C hits his draw on the first runout, then player B suddenly gets a chance on the second runout for his middle set to hold and thereby win half the pot.

To put some numbers to it:
Equity if everyone runs once is roughly A (top set) 60% B (middle set) 5% C (draw) 35%

Equity if B and C run twice while A runs once in the same main pot is now more like A 60% B 20% C 20%

I still stand by my comment that each individual pot (main & side/s) have to be run the same for all players invested in each individual pot, otherwise some players may get equity freeroll and EV advantage over others, which defeats the purpose of running a board multiple times which is to lower variance without changing the EV or all in equity for any player in any pot at any time.

Agreed. Doesn't A lose out on some fold equity, if main is RIO and side is RIT?

+1

Example for NLHE
Board: As Js 6d

Big Stack RIO: AA
Big Stack RIT: KsQs
Short Stack RIT: 66

Now if KsQs wins board 1, Shorty gets to join him on board 2 and go from almost dead to flipping for half.