We introduce a generalization of Goodman surgery to the category of projectively Anosov flows. This construction is performed along a knot that is simultaneously Legendrian and transverse for a supporting bi-contact structure. When the flow is Anosov, our operation generates the same flows of Goodman's construction. The Anosovity of the new flow is strictly connected to contact geometry. We use this relation to give a refinement to the sequence of surgeries coefficients producing Anosov flows

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... concrete examples. We give an interpretation of the bi-contact surgery in terms of contact-Legendrian surgery and admissible-inadmissible transverse surgery and we deduce some (hyper)tightness result for contact and transverse surgeries. Outside of the realm of Anosov flows, we produce new projectively Anosov flows on hyperbolic 3-manifolds with leaves of genus g>1 in the invariant foliations.