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#!/usr/bin/perl

...
push @parts, $& while / (Point\s[\d.]+:) .*? (?=Point|\z) (?!\1) /gsx;

use Data::Dump 'dd'; dd \@parts;

[
  "Point 1.3.4: A piece of text.\n\n",
...
  "Point 1.3.6: Another piece of text. Point 1.3.6: For some reason th
+is piece of text isn't finished yet.\n\nPoint 1.3.6: In fact, this pi
+ece of text even broke into a new line.\n\n",
  "Point 1.3.7: Finally, a new piece of text.\n\n",
]