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New variational principles are developed for the effective conductivity tensor for anisotropic two-phase electric conductors. Here the interface between phases is assumed to be highly conducting. Extra geometric information is encoded into the principles through the solution operators of simpler transport problems. These operators can be expressed as gradients of simple layer potentials with densities supported on phase interfaces or in terms of simple Dirichlet problems inside each phase region. New upper bounds on the effective conductivity are found that depend upon component volume fractions, a surface energy tensor and a scale-free matrix of parameters. This matrix corresponds to the effective conductivity tensor associated with the same geometry but with perfectly conducting inclusions. New lower bounds are given in terms of two-point correlation functions, component volume fractions, and interfacial …