I cycled in behind an odd bike today. The guy on it looked like a Courier (or reasonably plausibly dressed Fakenger). Initially I thought it was fixed, because it lacked a rear brake, but then he freewheeled.
The wheels were white plastic five spoked things, and this I think was the reason for no rear brake. The rear wheel wobbled rather horribly side to side (not unusual with plastic wheels), and it would have been hard to have fitted a brake which didn't rub. The frame looked recently painted, but only seemed to have fittings for a caliper brake, which made me think it was possibly not of a recent vintage.
It had quite big tyres, which was unusual, and wouldn't have helped fitting a caliper brake, although there seemed to be one on the front wheel. The tyres didn't seem to slow him down, he was doing 20mph+ along Queenstown Road, and at his cadence, it must have been geared 70"-ish, certainly not the lower gearing that Singlespeeds tend to be set to compared to most Fixedwheels.