Rumford established himself as the early leader in front of his home crowd with a six-under round of 66, while Foster matched that in the afternoon session as both players recorded rounds with just one bogey.
The pair are just one shot clear of the chasing pack, which contains 10 players on five-under after day one.
The Super 6 format sees a change in the usual golf tournament in that the top 24 players after 54 holes will contest a six-hole matchday event on Sunday to determine the champion.
"It's a brand new event for everyone so you break your mentalities. I'll have the same mentality tomorrow, and that is obviously to try and get in that 24," Foster told the European Tour website.
"But if you're not at your best and you travel this far, sometimes you're just quiet if you make the cut, but that's not good here.
"So I think that's maybe why you'll see better scoring, because everybody has got to aim for that highest spot and the higher people aim, results tend to get better."