For the two secret services to be working jointly on this project, they would more or less have to have been working together on many other things previously. It's not as if either side truly needed the assistance of the other to pull it off (although the Cardassians were more in need than the Romulans); the cooperation must have come from already existing good rapport rather than merely the pressing needs of the moment.
We learn in "Dax" that Terok Nor featured some components of Romulan origin. Industrial espionage or a secret technological pact? I'd actually wager the latter, as Romulans would be playing the Cardassians against the Klingons in that timeframe, during the Betreka conflict. Tal'Shiar and the Obsidian Order might have begun working together at that time in order to do more damage to the Klingons (even though of course they'd also do their best at backstabbing each other), and thus the channels would exist for Tain to relatively directly contact the Romulans and ask for their cloaks.
(And really, Tain would know what Starfleet knows, more or less, as Starfleet was on his turf. Nothing aboard DS9 could be expected to remain secret from the OO for long. The place must have been teeming with agents other than Garak, such as Bajorans enrolled during the Occupation, and with surveillance devices even Gul Dukat knew nothing of.)
Timo Saloniemi