It is a chrysler 2.2 with a garret turbo. It was a few years before the crapobishi calaboration.
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Here is some info from allpar.com:
The Daytona first debuted from Dodge in 1984 along with its look alike, the Chrysler Laser. Design was based mostly on the K-cars (Reliant and Aries), and retained the fuel efficient 2.2 liter 4 cylinder engine. The Daytona was mounted on an extended K-frame and had front wheel drive. The Daytona has always been a 3 door 4 seater, but the rear seats are very small and it should really be considered a 2 seater for all practical purposes, with its small 97" wheel base. These cars are known as ‘G’ cars because their body style was designated G (later, AG). Stock performance with the 2.2 was not up to the Daytona’s looks, but it was produced as an economy car and Dodge relied on the turbo versions and sporty looks to attract younger drivers; few automakers made a fast, affordable sporty car at the time. The Laser was aimed at older drivers who appreciated a plusher vehicle, and with luxury options. In both cases the five-speed stick-shift provided a substantail speed and gas mileage boost.
The car first debuted as the prototype G-24 Super Sports Car in 1982-83. If you look real close and squint a little you can see remnants of body styling from the 1969 Charger Daytona that swept NASCAR.
The original Daytona models had a length of 175 inches on a 97 inch wheelbase, a width of 69 inches, and a height of 50 inches.
In October 1982, Motor Trend wrote that the 1984 “Chrysler G-24” would have four bucket seats, be a hatchback, and stand on the K-car platform but share no body panels with any other Chrysler product. They noted the fuel-injected 2.2 as the base engine, saying there would be an optional turbo 2.2, but spent more time on the interior, “definitely sportier” than other Chrysler vehicles, with “restyled analog instruments and a few digital items.” (Thanks, NDNRacer)