How could Eisenheim achieve those reflections on the scene in that era?

Magical performances comprising of spirits being summoned on stage where pretty common even in the 19th century. The effect was achieved indeed with the use of smoke and mirror. However, a typical tool in usage was the Fantascope aka Magic Lantern. This apparatus is also seen in the movie but can be easily overlooked as normal lanterns.

I found this interesting piece of trivia on IMDB:

The method for creating the ghosts as shown to inspector Uhl involved the projection of a pre-recorded image into a hazy background. Since the ghosts Eisenheim conjured could speak to and interact with the audience, he most likely used a different method popular among magicians at that time. A fantascope was used to illuminate a real person off stage. The image was reflected off of a mirror or glassplate, creating a ghosted image. The lanterns that Eisenheim tells his assistants to leave behind when they are packing up the workshop bear a strong resemblance to fantascopes.