Maybe this package can help https://github.com/bachrathyd/MDBM.jl
Here is some glue code CalculusWithJulia.jl/implicit_equation.jl at master · jverzani/CalculusWithJulia.jl · GitHub
Otherwise, please see IntervalConstraintProgramming.jl/index.md at master · JuliaIntervals/IntervalConstraintProgramming.jl · GitHub which should also be more than up to the task