Eukaryotic genomes are packaged into a 3-dimensional structure in the nucleus. 
Current methods for studying genome-wide structure are based on proximity 
ligation. However, this approach can fail to detect known structures, such as 
interactions with nuclear bodies, because these DNA regions can be too far apart 
to directly ligate. Accordingly, our overall understanding of genome 
organization remains incomplete. Here, we develop split-pool recognition of 
interactions by tag extension (SPRITE), a method that enables genome-wide 
detection of higher-order interactions within the nucleus. Using SPRITE, we 
recapitulate known structures identified by proximity ligation and identify 
additional interactions occurring across larger distances, including two hubs of 
inter-chromosomal interactions that are arranged around the nucleolus and 
nuclear speckles. We show that a substantial fraction of the genome exhibits 
preferential organization relative to these nuclear bodies. Our results generate 
a global model whereby nuclear bodies act as inter-chromosomal hubs that shape 
the overall packaging of DNA in the nucleus.