Francine left Frobisher because she was a better detective than him. (COMIC: The Shape Shifter, AUDIO: The Maltese Penguin) She told him that he wasn't the Ogron she'd fallen in love with. (AUDIO: The Holy Terror)
Francine used the disguise of Arthur Gringax. She produced a product that gained the attention of Josiah W. Dogbolter. The product was a microchip containing a joke.
Francine assumed the name Alicia Mullholland. She went to Frobisher and hired him to seek out Arthur Gringax. She morphed back into Arthur and pretended to be dead, allowing Frobisher to find her. Frobisher later found her in his office. She professed her love for him but he refused her. They were then confronted by Elric Chandler, who threatened them. Francine told Chandler that he was likely to be killed, and he was; a smart bomb detonated, killing him. They were taken to Dogbolter where his curiosity made him use the microchip. The joke gave the workers imagination, ruining the economy. This freed them from Dogbolter's oppression.
Francine then revealed her true identity to Frobisher. She told him that he'd regret staying with her and not leaving in the TARDIS, and that she wanted to bring Dogbolter to justice. (AUDIO: The Maltese Penguin)