A formlet at any time can be in an accepting or rejecting state. A dependent formlet can only execute if the formlet it depends on is in an accepting state, e.g. it has produced a value. So the trick is to propagate some value instead of a failing state.

You can produce a value and indicate the lack of an inner value with several techniques. The most intuitive one is wrapping your formlet result in an option value. For instance, here is a dependent formlet whose second input box is "non-blocking":

    [<JavaScript>]
    let Form () =
        let F () =
            Controls.Input ""
            |> Validator.IsNotEmpty "Must specify a value"
        Formlet.Do {
            let! _ = F ()
            let! _ =
                F ()
                |> Formlet.MapResult (fun res ->
                    match res with
                    | Result.Failure _ ->
                        Result.Success None
                    | Result.Success v ->
                        Result.Success <| Some v
                )
            let! _ = F ()
            return ()
        }