The duo, brothers Matt and Luke Goss, were due to play a full UK tour this summer, after a break of 24 years.
But now dates in Newcastle, Glasgow, Nottingham and Birmingham have been scrapped.
A statement said the dates had been cancelled "due to unforeseen logistical circumstances", with the brothers offering "sincere apologies" to fans.
Two shows at London's O2 arena will go ahead, as will a date at the Manchester arena on 22 August.
The first London date, on 19 August, marks the anniversary of their farewell show at Wembley Stadium in 1989.
It sold out in seven seconds. However, there are several hundred tickets available for the remaining shows.
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