Which? claims the ads are "misleading and irresponsible" because the chocolate-lined wafer tubes are being promoted "as a way to get children to drink more milk".
The straws, launched last month, are aimed at children who shun other cereals. The ads use the line "A totally different kind of breakfast" and suggest the product makes drinking milk more fun.
"It sends a confusing message about what is healthy," said Which? chief policy adviser Sue Davies. "The implication that the chocolate straw is wholesome if consumed in this way is visually and verbally unclear."
The complaint is the first the ASA has received about the ads.
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