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Multiple realities
(covers information from several alternate timelines)

The following is a list of unnamed 23rd century Romulan Birds-of-Prey.

By commander

By encounter

Narada stowaways

These three Birds-of-Prey were aboard the Narada in an alternate 2258. Spock flew past these ships as he escaped the Romulan mining vessel aboard the Jellyfish. (Star Trek)

Praetor's flagship

In 2266, the Praetor's flagship was involved in the Neutral Zone Incursion with the Federation. The ship attacked several border outposts, destroying them utterly. The USS Enterprise responded to the alert and engaged in a tense game of cat-and-mouse with the intruder for more than ten hours, before the Bird-of-Prey was destroyed. (TOS: "Balance of Terror")

The reference book Star Trek: Federation - The First 150 Years identifies the name of the Praetor's Bird-of-Prey as the Algeron.
The Star Trek Customizable Card Game gives this ship's name as Gal Gath'thong (β), named after the Gal Gath'thong mountain range on Romulus. The Decipher RPG sourcebook Starships calls this vessel the Tellus while The Way of D'era calls it the Nhorazz. The more recent Star Trek Adventures - Beta Quadrant calls it the Talon.
The Starships RPG sourcebook gives its registry as IRC 1100.

Neutral Zone patrols

Three vessels pursue the Enterprise

In 2267, ten Birds-of-Prey engaged the USS Enterprise after it entered the Romulan Neutral Zone. (TOS: "The Deadly Years")

The original footage of these vessels was recycled from "Balance of Terror".

Capture of the Enterprise

Two Romulan D7s and a Bird-of-Prey surround the Enterprise

In 2268, this Bird-of-Prey, along with Tal's flagship and a second vessel participated in the capture of the USS Enterprise after it crossed Romulan Neutral Zone.

They were eventually outmaneuvered by the Enterprise after it engaged a cloaking device stolen from the Romulan flagship. (TOS: "The Enterprise Incident")

Original appearance

In the original broadcast, all three Romulan vessels were D7-class and were represented by the same studio model. One of the battle cruisers was replaced with a Bird-of-Prey in the remastered version of the episode.

Vulcan museum piece

Romulan Bird-of-Prey, 2380.jpg

This Bird-of-Prey was on display as a museum exhibit on Vulcan until it was stolen by Starfleet officers Andy Billups, Shaxs, and Sam Rutherford. The vessel was then used by Jack Ransom, D'Vana Tendi, and three other members of a black ops team to infiltrate the Romulan Neutral Zone and arrive at Romulus to perform an elaborate rescue of the Prime Imperium Magistrate, Clar. (LD: "Veritas")