A　biological　vesicle　in　fluid　environment　is　described　by　a　conservative　Allen–Cahn　type　phase-field　model　and　the　incompressible　Navier–Stokes　equations.　To　accurately　and　efficiently　solve　this　complex　system,　we　present　a　totally　decoupled,　linear,　and　second-order　time-accurate　method　based　on　a　time-dependent　auxiliary　variable　methodology.　The　time-discretized　versions　of　energy　stability　and　unique　solvability　are　analytically　proved.　By　using　a　simple　and　effective　energy　correction　technique,　the　consistency　between　the　original　and　modified　energies　is　enhanced.　The　proposed　numerical　algorithm　is　simple　to　implement　because　we　only　need　to　separately　solve　linear　elliptic　equations.　Various　computational　tests　are　conducted　to　verify　the　performance　of　the　proposed　numerical　algorithm.　©　2022　Elsevier　B.V.