I'd just do a standard merge but throw away the results as you go unless the next output is 1+previous.

#!/usr/bin/perl -w
use strict;

my @list= ( # A more interesting test:
    [ 101, 105, 204, 312 ],
    [ 100, 313, 409 ],
    [ 205, 206, 211, 315 ],
    [ 106, 207, 210, 314 ],
);
# Temporarily get list numbers sorted by first item
my @next= sort { $list[$a][0] <=> $list[$b][0] } 0..$#list;

# $p is the index of the list to pick from next
# $next[$p] says which list to pick from after list $p
( my $p, @next[@next] )= ( @next, -1 );

my @run;    # Our consecutive run so far
my @from;   # Which list we selected each of the above from
my @best= ( undef ) x 2;    # The run to beat

while(  1  ) {
    # Add smallest remaining item to our run
    push @run, shift @{ $list[$p] };
    push @from, $p;

    # If we've run out of items or if this one isn't consecutive
    if(     ! defined( $run[-1] )
        ||      1 < @run
            &&  $run[-2]+1 < $run[-1]
    ) {
        # then see if the now-finished run is a winner
        pop @from;
        if(  @best <= @from  ) {
            # We've at least tied, so report
            if(  @best < @from  ) {
                # A new size reached!
                print "$#run-int runs:$/";
            }
            @best= @run;
            pop @best;
            print "$best[0] .. $best[-1] ( @from )$/";
        }
        last   if  ! defined $run[-1];  # We ran out of items
        # The next run starts with the item we just added
        @run= $run[-1];
        @from= $p;
    } elsif(  1 < @run  &&  $run[-1] <= $run[-2]  ) {
        # The lists weren't sorted or our code is broken
        die "Abort; out of sequence: @run$/";
    }

    # Move $list[$p] now that it has a bigger first item
    my $n= $next[$p];
    if(  ! @{ $list[$p] }  ) {
        # Empty.  Don't insert $list[$p].
        $next[$p]= -2;  # Not really needed
        $p= $n;
    } elsif(    $n < 0
            ||  $list[$p][0] < $list[$n][0]
    ) {
        # Do $list[$p] again
    } else {
        my $x= $n;
        my $y= $next[$x];
        while(  0 <= $y
            &&  $list[$y][0] < $list[$p][0]
        ) {
            $y= $next[ $x= $y ];
        }
        # Do list[$p] after $list[$x], before $list[$y]:
        $next[$x]= $p;
        $next[$p]= $y;
        $p= $n;
    }
}
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