What rules of English allow the first t in “patient” to make an sh sound?

What makes you think it is a rule?

Languages are not designed to a set of rules laid down by a designer. That idea was tried with Esperanto, and it doesn't work.

Language is organic. It grows and shifts with the tides of time and change.

English is an evolving entity, and many of its seemingly strange rules, only exist in the wishful mind of non-native speakers.

I don't know of any native speaker who has ever thought much about Engiish spelling, other than "I am good at spelling", or "I am not very good at spelling".
What many non-native English speakers fail to realize is that English is English. It has not requirement to behave as they may expect! and it doesn't!

Is English spelling hard for a non-native speaker? Yes, but that is just the way it is!

Is English spelling hard for a native speaker? Not especially, but it is a learnt art; learnt over many years.

Many of the quirky spellings come from the pronunciations of the French, German, Latin, Gaelic, etc. root languages... and believe it or not, there is an underlying rhythm; a series of patterns based somehow on the different root languages.

What is the rule? you ask... You have asked the wrong question.

Perhaps a better question is how do I deal with all the unusual spellings?
The answer is: Remember them. Remember each and every one of them!

If you want to learn English, well that is how every native English speaker has learnt them; by memorizing them...
There is fundamentally no other way..
The spelling of words is not an automatic process.. I had to learn each and every word. All native English speakers had to learn each and every word.

It is simply a parallel discipline! ... jus like speaking is discipline.

As as for all that utter nonsense about ghoti, it is an absolute waste of time! You are missing the point, and wasting your time with what non-native speakers see as conflicting spellings.
There is no conflict! ... because the primary rule of spelling English words is MEMORY!

Slowly, slowly, the patterns reveal themselves.. (even if it is unconsciously)

PS: Mathew.. I hadn't noticed that you were in the US... I'm so used to dealing with non-native speakers in language-exchange environments, where many seem to think that Engish spelling is based on a few simple logical rules... Oh well, my answer may be more appropriate for ESL readers of this question..

The reason something pronounced [ʃ] ("sh") would ever get the spelling "ti" is because of palatalization. Basically, the "io" diphthong contains a palatal consonant [j] ("y" sound), which, in certain cases, pulls the place of articulation of other consonants towards it (e.g. t->ʃ).

The palatalization process is no longer active in these words spelled with "ti" (nowadays it is just 100% /ʃ/), but palatalization is active in some British dialects, for pronunciations of words like "Tuesday": /tjuzdeɪ/ ("Tyoozday") comes out as [tʃuzdeɪ] ("Choozday").

(Note that in most dialects of American English, we just say [tuzdeɪ] ("toozday"), because we don't have a palatal consonant in that context at all.)