Very interesting comment by Dr. Jeff Master's at Wunderground. Because I am in Houston, I have bolded the section regarding Allison
:Formerly, Eastern Pacific storms crossing into the Atlantic would be given a new name, but a recent NHC policy change allows storms to keep their names when they cross from one ocean basin to another. If Barbara were to dissipate before reaching the Gulf, then its remnants regenerate into a tropical storm in the Gulf, it would be named Andrea.
Double ocean tropical cyclones: a rare breed
According to the Hurricane FAQ, since 1923 there have been four East Pacific tropical storms or hurricanes that have maintained their circulations while crossing into the Atlantic Ocean, becoming tropical storms in the Atlantic Ocean:
Northeast Pacific Tropical Storm Alma (May 2008) became a remnant low in the Atlantic, where it merged with another tropical wave which generated Atlantic Tropical Storm Arthur. Arthur hit Belize as a tropical storm, killing nine and doing $78 million in damage.
Northeast Pacific Hurricane Cosme became Atlantic Tropical Storm Allison in June 1989. Allison hit Texas as a tropical storm, and heavy rains from Allison--up to 30" in some regions of Texas and Louisiana--triggered floods that killed eleven people and did $1 billion in damage. (A later incarnation of Tropical Storm Allison in 2001 also hit Texas, and caused such extensive flooding that its name was retired.)
An unnamed Northeast Pacific Tropical Storm (September-October 1949) became Atlantic Hurricane (Storm #10) and hit Freeport, Texas as a Category 4 hurricane, killing two people.
An unnamed Northeast Pacific Tropical Storm (October 1923) became Atlantic Hurricane (Storm #6) and made landfall as a Category 1 hurricane in Louisiana.
There have been eight Atlantic tropical storms or hurricanes that have maintained their circulations while crossing into the East Pacific Ocean, and were then tropical storms in that ocean.