В холодильнике ничего не осталось (поесть)?
With ''поесть'' (perfective) this particular sentence sounds very natural, whereas with ''есть'' (imperfective) - pretty weird.
At the same time, it is perfectly natural to say, ''Есть больше нечего.''
I understand that for non-native speakers, this may be very confusing in terms of rational logic.
In order to confidently make choices in such expressions, you have to develop the ''Russian logic'' - associative thinking based on analogies to many Russian idioms, colloquial expressions, and commonly used everyday phrases. It is very easy to make mistakes if you try to deduce the right choices by purely logical, rational reasoning. For native Russian speakers it is usually much easier to tell the right choice than to try to find the logical reason as to why it is the right choice. Native Russian speakers make their choices based on the subconscious associative ''Russian logic'' rather than on rational reasoning. The correct variants are simply those that are easier perceived by the Russian-speaking mind. Words and expressions in the Russian language activate specific thinking patterns and emotions that are often not easy to describe logically and rationally in another language. You must ''feel'' the language and its specific words and expressions on the subconscious level in order to speak naturally.
So let me answer your question from the ''Russian logic'' standpoint. This logic may be different from what you find in your grammar textbooks, but is what how Russian speakers actually think.
Let's start with the following phrase: ''Что ему осталось сделать?'' This sentence implies that the person has a checklist and has already done some things on that list, so the question is about what is left on the list. Both verbs here are perfective, so the sentence sounds very result-oriented, and this is why it implies a checklist with clearly defined goals.
Let's now make both verbs imperfective: ''Что ему остается делать?'' Now the most natural perception is that the person is in a difficult situation in which he has very few options left, and the question is about exactly what options are left. The question sounds somewhat global, strategic. In the past tense, the same question takes the following form: ''Что ему оставалось делать?'' The latter sentence is often used as an apology for someone's actions or explanation of someone's actions by saying that he had no other choice.
The variant with the perfective verbs sounds result-oriented, while the variant with the imperfective verbs - process-oriented, hence the difference in the perception.
Overall, perfective and imperfective verbs activate different thinking patterns - result-oriented vs process-oriented, so if you use perfective and imperfective verbs together in a single sentence, your conversation partner may get somewhat confused on the subconscious level, depending on the context.
I want to illustrate this by how many results Google gives for the following combinations:
"Что ему осталось сделать?" (perfective + perfective): 9,000 results
"Что ему остается делать?"(imperfective + imperfective): 24,000 results
"Что ему оставалось делать?" (imperfective + imperfective): 53,000 results
"Что ему осталось делать?" (perfective + imperfective): 1,000 results
"Что ему остается сделать?" (imperfective + perfective): 5,000 results
As you see, the pure forms (i.e., ''perfective + perfective'' and ''imperfective + imperfective'') are more common than the mixed ones, although the latter are not ungrammatical. The pure forms are just easier for perceive and are thus preferred, although in some contexts you may need to use a mixed form in order to emphasise some nuance (as I will explain below).
Now back to your refrigerator.
Let's make two obvious observations:
(i) The verb ''есть'' (in the meaning ''to eat'') is perceived as a general concept or a process, whereas ''поесть'' - rather as ''to have a certain meal at a certain time (e.g. now).'' ''Поесть'' is a result-oriented word and denotes a limited action. ''Есть'' is a process-oriented word and feels rather global.
(ii) The verb ''осталось'' is a result-oriented verb. ''Ничего не осталось'' is perceived as a result of a completed action.
''В холодильнике ничего не осталось поесть'' sounds perfectly natural. It is a pure form (perfective + perfective); the idea is expressed purely in terms of result-oriented thinking; the phrase is about particular circumstances rather than a global state of affairs - after all, you can always buy food in a shop to resolve the issue. The perception is, ''Nothing has been left in the fridge to quickly have a meal right now.''
''В холодильнике ничего не осталось есть'', in contract, triggers a weird interplay between result-oriented thinking and process-oriented thinking in the Russian-speaking mind. It is perceived as weird as, ''As a result of some particular process, no options have been left in the fridge as to what to eat globally.''
Just another example: ''Решил пойти купить покушать'' is strongly preferred to ''решил пойти купить кушать,'' for the same reason.
Now let's consider this: ''Есть больше нечего.'' It sounds natural. The idea is expressed purely in terms of global, process-oriented thinking. If you now replace ''есть'' by ''поесть'', it will sound unnatural, because ''больше'' is related to a process (namely, the process of gradually consuming food), while ''поесть'' is a result-oriented word and denotes an isolated action. In some rare contexts, however, ''поесть больше нечего'' sounds okay (e.g., the person ''поел'' a few times and asks whether there is something to ''поесть'' again).
Note that mixed forms (i.e. combinations of perfective and imperfective verbs) are okay to use if the context and the intended meaning require it.
I can even give you an example where ''ничего не осталось есть'' sounds fine: ''Пожар уничтожил запасы хлеба, и жителям ничего не осталось есть.'' (''The fire annihilated the reserves of bread, so the population was left with nothing to eat''). Here it is emphasised that the population was left with nothing to eat globally, not just to have a meal right now. This is why ''ничего не осталось есть'' sounds natural in this context, as opposed to the context with the fridge.
To conclude, the word choice is dictated by the context, the intended meaning, and the way how the words and their combinations are perceived by the Russian-speaking mind.
