And here's one that also allows left angle bracket to be escaped... 

#!/usr/bin/perl

# https://perlmonks.org/?node_id=1222687

use strict;
use warnings;
#use hairy_regex;

my $nobacktick = qr/(?:(?:\\.)++|[^\\`]++)*+/s;
my $noleftangle = qr/(?:(?:\\.)++|[^\\<]++)*+/s;

while( <DATA> )
  {
  s/ \G $nobacktick \K ` $nobacktick ` /
    $& =~ s! \G $noleftangle \K < !&lt;!gxr /gex;
  print;
  }

__DATA__
This is `my <string>` that I want to modify because the angle bracket 
+is between backticks
This is `my &lt;string>` that I want to modify because the angle brack
+et is between backticks
This is `a <string> I don't want to modify because it's not between ba
+ckticks
This is \`another <left_angle_bracket>`I don't want to modify because 
+the first backtick is escaped but I do want to modify <the_last_left_
+angle_bracket>` between the backticks
`whole string with <string> enclosed in backticks`
`whole string with <string> not enclosed in backticks
`whole string with <string> not enclosed in backticks\`
``whole string with <string> not enclosed in backticks`
`whole string with \<string> enclosed in backticks`

[download]

Outputs: 

This is `my &lt;string>` that I want to modify because the angle brack
+et is between backticks
This is `my &lt;string>` that I want to modify because the angle brack
+et is between backticks
This is `a <string> I don't want to modify because it's not between ba
+ckticks
This is \`another <left_angle_bracket>`I don't want to modify because 
+the first backtick is escaped but I do want to modify &lt;the_last_le
+ft_angle_bracket>` between the backticks
`whole string with &lt;string> enclosed in backticks`
`whole string with <string> not enclosed in backticks
`whole string with <string> not enclosed in backticks\`
``whole string with <string> not enclosed in backticks`
`whole string with \<string> enclosed in backticks`

[download]

(more test cases probably required)

In reply to Re: Replacing left angle bracket with HTML entity when between two backtick characters by tybalt89
in thread Replacing left angle bracket with HTML entity when between two backtick characters by nysus

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