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We provide a routing framework for hybrid RF/FSO backbone networks, utilizing the characteristics of RF and free space optical (FSO) links. FSO links offer higher bandwidth and security, while RF links offer more reliability. We propose the concept of criticality index for different classes of traffic and providing obscuration-tolerant paths to the traffic in a weighted max-min fair way. We provide an optimal algorithm for the case where a traffic demand can be routed along multiple paths. The problem of routing unsplittable traffic is NP-hard, so we propose efficient heuristics for routing them. We do extensive simulations to demonstrate that our algorithms outperform the algorithms currently in use.