Thanks for your thoughts, Hieronymus. We do not belong to any actual parish — instead we have a couple of visiting diocesan priests who celebrate the Tridentine Mass in a chapel nearby (south-central WA). Once an EWTN priest told us by phone we were perfectly OK in our present circumstances to receive the Holy Eucharist after Confession like any other Catholics. But I have always wondered if that was correct. It is difficult to catch up to the traveling priests to discuss this but I will have to try harder. Another possibility is talking to priests at an FSSP parish one state over. I don’t even care about having the marriage “straightened out” since we are as brother and sister anyway, just want to receive Holy Communion. Same with my wife.
As you intuited earlier, there are reasons for having the marriage “straightened out” that go beyond the realm of not living as brother and sister—all things being equal, a Josephite marriage is/can be a good thing.
Your wondering if the advice is correct is a good thing—you don’t need to make your Holy Communion specifically after Confession any more than any other Catholic. Either you had a firm purpose of amendment in the past and are in a morally upright situation, or you are not in a morally upright situation and need the situation amended before absolution is given. The main wrinkle is the appearance of scandal—but that largely depends on how widely known your initial marital state is—which is why going somewhere anonymously is sometimes suggested as an option. FSSP is likely good.
“Straightening it out” wouldn't be very difficult. As baptized Catholics, your marriage lacks proper form. It can be regularlized with little effort.
If for no other reason than to put to rest any questions in your mind, you might want to talk to someone in your diocese.
Or go talk to the FSSP folks. I don't think they'll steer you wrong.
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