It may seem difficult to find something that ties an antique Chinese console with an African mask with a Persian rug -- but it's doable. And it's necessary if you want to create a harmoniously eclectic look.
Just a few potential tie-ins include:
Theme -- Do they recall a similar event, lifestyle or region? Are they all calming, or festive or ancient?
Color -- Don't go for pieces that are all red or all blue or all yellow, but instead, look for similar color schemes, a common color that runs through each style, or simply a common "temperature" -- for instance, red and orange are warm, while blue and purple are cool.
Scale -- Basically, this refers to size, or grandness, and for your main styles, you want this to be a shared trait. A massive, stone coffee table from Santa Fe will dwarf the tiny, hand-painted cabinet you found at the French antique store. Balance should be a prime consideration when combining disparate elements.
Finally, even as you practice restraint, don't forget why you started this project in the first place: Pairing unexpected styles can be so much fun …