What kind of tense is this?

The verb would is a modal verb used to express willingness in the past, polite request, preferences/desires, offers and invitations. A modal verb is a type of auxiliary (helping) verb that has no meaning on its own but modifies the main verb, changes its meaning and gives more detail about the action. In the way you have expressed it though, you are using the present or future tense to discuss a potential event, should it occur.

Its structure, in the affirmative form, is: Subject (I) + would (would) + verb (do)+ …

The only difference between this structure and the phrase that you have given is that yours adds the phrase 'to have to' before the main verb because you are expressing obligation.

Other examples in the future/present:

• I would go abroad immediately
• He would have to sell his house if the business went bankrupt.
• They would need extra provisions, if an earthquake were to occur.

Technically, the word "would" is in the past tense. But that's a really bad way of thinking about it.

In reality, this is a second conditional with the "if" clause omitted. That means that the verb is in the past tense of the subjunctive mood, but the phrase as a whole expresses a hypothetical event in the present or future.

Sadly, English conditionals often break the time period specified by their verb's tense. In fact, I didn't know that "would" was the past form of "will" until I answered this question--I just thought of "would" as the form you use in conditionals.