Foster and Stavrou, who is the holder of the women's world matchplay singles, made it through on a tie break against Welsh duo Kerry Packwood and Jason Greenslade.
Foster said "The first set was amazing, you could have put a plate over some of the heads. However the second wasn't just quite as inch perfect. There was some good bowls played but we lost. We put that to the back of our minds and concentrated on the tie break ends and got there in the end and that's all the matters".
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Hide AdDavies, who won the title for the second time last year when she was six months pregnant, and David Gourlay, going for a recored third title in a row, also needed a tire break to see off Israel's Ruthy Gilor and Andy Thomson.
In today's other quarter-finals at Potter's in Norfolk, East Lothian star Alex Marshall partners Carol Ashby, the former world champion from Eastbourne who now represents Midlothian, against England's Sarah Seymour and Greg Harlow. England's Mark Royal and Dee McSparran from the USA take on the England/Guernsey combination of Mervyn King and Alison Merrion in the other half of the draw.