Quoted via Wikipedia:
Time Gal[a] is an interactive movie video game developed and published by Taito and Toei Company, and originally released as a laserdisc game in Japan for the arcades in 1985. It is an action game which uses full motion video (FMV) to display the on-screen action. The player must correctly choose the on-screen character's actions to progress the story. The pre-recorded animation for the game was produced by Toei Animation.
The game is set in a fictional future where time travel is possible. The protagonist, Reika, travels to different time periods in search of a criminal, Luda, from her time. After successfully tracking down Luda, Reika prevents his plans to alter the past. Time Gal was inspired by the success of earlier laserdisc video games that used pre-recorded animation, including Dragon's Lair (1983) and the previous Taito/Toei collaboration Ninja Hayate (1984), while Reika's character design bears similarities to the anime characters Lum (from Urusei Yatsura) and Yuri (from Dirty Pair).
The game was later ported to the Sega CD for a worldwide release, and also to the LaserActive in Japan. The Sega CD version received a generally favorable reception from critics.
Why Time Gal? Well, I own exactly two SegaCD games. Guess what one of them is?
Personally, I've always enjoyed these types of games.