Long story short
Consequenced has been in use for nigh on 200 years, albeit not exactly at the top of the pile of first words to use when you want to talk about being on the receiving end of consequences so it seems that to call it an error would be something of an injustice; without a governing language body determining what's right and what's wrong in English; unusual but becoming more common might be a better term. (Someone must have asked for a SWR to cover that).
Despite OED having a mention of consequence as a verb it's quite likely that the recent popularisation doesn't come from Milton's Tetrachordon given the gap between 1645 and 1830 where I could find no mention, it could well be that consequence has been 're-verbified' although it's quite possible it was in use but there are just no scanned documents for that period. But from 1830 onward, there's no doubt that consequenced is being used mainly to mean having consequences applied to a person or thing.
OED
II. ˈconsequence, v.Obs.rare—1 [f. prec. n.]
intr. To draw inferences or conclusions.
1645 Milton Tetrach. (1851) 175 Such a methodical and School-like way of defining, and consequencing.
There is a more recent dictionary entry, which simply echoes OED, but is not marked obsolete or even rare.
Websters Unabridged Dictionary (1913)
(n.) Consequencing : Drawing inference. [R] Milton.
I'm not sure how OED/Webster got drawing inference/conclusions from Milton's work - I think he meant exactly what it says on the tin adding consequences to an action (marriage/divorce). There's a lengthy chunk about that at the bottom of the answer.
Long story
One of the first Google results I get is this analysis of that moment on the show.
The Sopranos on the Couch - Analyzing Televisions's Greatest Series (Maurice Yacowar - 2003)
Google Books
During his meeting with Father Hagy and school psychologist Dr. Galati, we share Tony's skepticism about the new jargon ("Anthony has misbehaved. He should be consequenced"). Tony rejects the school psychologist's report because he considers AJ a 13-year-old boy not a case. The specialists turn every problem "into a disease." Tony is frustrated that modish "consequencing" does not include the time honored "whack upside the head" or the traditional tarantelle on the bottom. His own father's favorite "child development tool" was the belt.
The word 'modish' is a bit of a giveaway despite modish also having an archaistic taint as the OED suggests
1.1 According to the mode or prevailing fashion. (Very common in 17–18 c.; now somewhat arch.)
Just how modish is it?
The Brigham Young Google Books query tool is vastly superior to anything Google provide for querying their corpora and in the Google Books (American) corpora it lists occurrences per decade of consequenced as
1830-5
1890-1
1920-3
1960-1
1970-1
1980-15
1990-19
2000-20
Of the 5 from 1830 - two are bad scans and one is a duplicate giving us just 2
Artemas Baker - 1832 - Circular, to the Honorable the Representatives of the People of Ohio in the Congress of the United States ... Google Books
I shall remark further, that, as the reducing the grade, was so important a part of his instruction; the neglect of it likely to be attended with such extensive ill consequenced; and one about which, the Engineer department could not be ignorant; ...
John Randolph - 1834 - Letters of John Randolph, to a Young Relative: Embracing a Series of Years, from Early Youth, to Mature Manhood Google Books
My horsemanship was, indeed, put into requisition, on meeting a rattley hacky coach, with lights, driving at a furious rate. It was where "the Avenue" is crossed by a gutter, and impeded by ice. Nevertheless, I did what Cambey could not do with his wretched curb-bridle - and as Simon says "I consequenced her with a snapper."
From the 1840's (UK publication)
John Goodwin - 1840 - Redemption redeemed: wherein the most glorious Work of the Redemption of the World by Jesus Christ, is vindicated against the Encroachments of later Times Google Books
For the more secure a sinner may be that his sinful practice will not be so sadly consequenced, as the nature and property of it only considered, it might very possibly be, the greater tentation lieth upon him to adventure upon it.
From the 1890's
Eugene F. Starke, Wilson A. Smith, Wesley A. Dunn - 1892 - The Medical Current, Volume 8 Google Books
Impaired nutrition, caused by weakened digestion, consequenced of previous disease, rapid growth, or any derangement if allowed to continue, naturally will end in impoverished Therapeutic Value of Remedial Gymnastics
Of the three from the 1920's one is a reprint of Randolph, one is a bad scan but one is valid.
Massachusetts. Supreme Judicial Court - 1920 - Report of Cases Under the Workmen's Compensation Act Determined on Appeal by the Supreme Judicial Court Google Books
The elevator at the time of his death was at the top of a building, and in some unknown manner the doors of the elevator were opened so that it was possible for him to have received the fall which consequenced his death.
The single 1960 hit actually provides 20 different books - because Google Books itself doesn't restrict it's list to any specific corpora. What is noticable is that most of those books that have consequenced properly scanned (and enough of a snippet to be able to determine the meaning) are from Indian sources. The obviously US scan shows me a snippet that is cut off right through the word consequenced but from the search page the text reads:
Iowa. Division of Historical Museum and Archives, State Historical Society of Iowa, Iowa. Historical Dept., Pub, Iowa. Historical, Memorial, and Art Dept - 1961 - Annals of Iowa Google Books
Such then confessedly is the very climax of deadly consequenced flowing from writing, printing, selling and reading works of fiction. Self-murder, and novel-reading as the cause! Who wrote the novel that did the deed of death by drowning?
The 1970 list gives another 20 or so books, many of them Indian, one Australian using ill-consequenced acts and number of legal type references to consequenced tax. But two are directly related to the Soprano use (even though one is Canadian - that's still North American)
Canadian Psychological Association - 1975 - Canadian Journal of Behavioural Sciences, Volume 7 Google Books (two mentions)
(1) This graph counts the number of times an individual target behaviour is consequenced with a 24-hour period. The data, consisting of all the fines and rewards given by staff for that target, are obtained from the boy's point card.
(2) A type "a" histogram seems to occur in one of two ways, either the youth actually has a low frequency of behaviour, or else some behaviours are not picked up by staff and consequenced.
(To me this is the first sign of consequenced being used both as punishment and reward, whereas I would normally associate consequences as being negative.)
John Muir - 1973 - The velvet monkey wrench Google Books
If your reactions test out so badly that the machine operator cannot, in conscience, give you any color of card at all, you'd better not drive. If you are Challenged, you could be Consequenced heavily by the People who would, for sure, see you in living color.
(I have no idea what the odd capitalization is about)
The 1980's are supposed to have fifteen US books, Google Books offers me 45; again many of them are from India, some from Australia, one from Pakistan, and another using the word/term ill-consequenced.
[Author Unknown] - 1983 - Journal of behavior therapy and experimental psychiatry, Volume 14 Google Books
Basically, antisocial behaviors, such as physical aggression, verbal abuse, and persistent disobeying, are consequenced with response costs and/or time-out. Prosocial behavior is reinforced through a modified DRO format.
Daniel D. Roman, Joseph F. Puett - 1983 - International business and technological innovation Google Books
The decision process to develop international markets has to factor in potential negative aspects consequenced by technological side effect, social pressures, ...
The 1990's should be nineteen US books, but again I'm presented with forty-five or so and again a good number of them are Indian and Australian in origin but it's clear that the general trend, of the US books, is turning towards behavioural issues and discipline.
Rob Kerr - 1995 - Stop and Think: Empowering Students to Manage Behavior Google Books
More and more "overreacting" students are counseled and consequenced in the homeroom. We see this as a good evolution. We now consider nullifying the SAT center altogether.
T. Steuart Watson, Frank M. Gresham - 1998 - Handbook of Child Behavior Therapy Google Books
First, given that the child is given work which is appropriately difficult and that this work is monitored and consequenced, is the child able to perform in the classroom? Second, is there a collateral decrease in disruptive behavior, or are there behaviors which will require some specific attention?
Foster Cline - 1994 - Conscienceless Acts, Societal Mayhem: Uncontrollable, Unreachable Youth and Today's Desensitized World Google Books
Osborn was taken to court for alleged child abuse after her adopted daughter, Sophia, attracted the sympathy of kindly passersby when she was being consequenced with techniques Sharon had learned in classes for parents with difficult children.
Note that a number of other books in this group print the word consequenced in 'quotes', perhaps to denote that it's use is not all that common (even though it actually was fairly common).
In the 2000's list there are over a hundred books listed and at least half of them are related to behavioural issues particularly relating to children's behaviour.
All in all, consequenced has been kicking about for a while, it's just not been very popular. From 1970 it's picked up considerable use in a way that seems to be a euphamism of sorts for punishment and discipline but also with the added 'edge' that consequences can also be rewards. It's not restricted to the US, a lot of Indian, Australian and more recently European published books feature consequenced.
Milton, OED & Webster
Some quoted parts from John Milton - 1645 - Tetrachordon Google Books
I'm not sure why OED and Webster define consequencing as drawing inferences [or conclusions].
In the full context of Milton's writing (he is discussing marriage & divorce) the rules of marriage (and consequences of breaking them) are not bound by "Religion, Law or Reason" but by "the meetness of help and solace, which is the formal cause and end of that definition that sustains them."
And he goes on to say
"... yet Moses, as if foreseeing the miserable work that mans ignorance and pusillanimity would make in the matrimonious business and endeavouring his utmost to prevent it, condescends in this place to such a methodical and School-like way of defining, and consequencing, as in no place of the whole Law more."
Those 'School-like' definitions are presumably from Matthew 5 & 19, Deuteronomy 24 and/or Corinthians 7 (Both include 'rules' of marriage and consequences of breaking those rules). Consequencing, to me, means adding consequences to an action in this case, marriage and divorce.
One should note that the Tetrachordon was only one of four pamphlets Milton wrote arguing for reformation of the laws regarding divorce in England and at the time, Milton was trying to get divorced from his wife.
Best Answer
It sounds to me like people are just leaving out words, as opposed to completely violating the grammar. This often happens in conversation.
It seems like people are just leaving out the helping verbs. It's certainly not proper but it's acceptable in casual conversation.
As for whether or not Miamians speak this way, I can't say, but I'd guess lots of native English speakers will speak this way from time to time. I'm in Canada and I would not find any of this speech surprising or difficult.