Recent studies have shown that collaborative intelligence (CI) is a promising framework for deployment of Artificial Intelligence (AI)-based services on mobile devices. In CI, a deep neural network is split between the mobile device and the cloud. Deep features obtained at the mobile are compressed and transferred to the cloud to complete the inference. So far, the methods in the literature focused on transferring a single deep feature tensor from the mobile to the cloud. Such methods are not applicable to some recent, high-performance networks with multiple branches and skip connections. In this paper, we propose the first bit allocation method for multistream, multi-task CI. We first establish a model for the joint distortion of the multiple tasks as a function of the bit rates assigned to different deep feature tensors. Then, using the proposed model, we solve the rate-distortion optimization problem under a total rate constraint to obtain the best rate allocation among the tensors to be transferred. Experimental results illustrate the efficacy of the proposed scheme compared to several alternative bit allocation methods.