Can an arcane trickster/wizard multiclass use an arcane focus for their arcane trickster spells?

Question

By default, an arcane trickster cannot use an arcane focus to cast their spells, as they lack the class feature that wizards and warlocks have that allows them to; this forces them to use a component pouch for their material needs.

However, as a wizard "can use an arcane focus as a casting focus for their wizard spells" and the spells arcane tricksters learn are repeatedly referred to as "wizard spells", can an arcane trickster/wizard multiclass use an arcane focus for the spells which they learnt from being an arcane trickster?

It may be useful to note that both arcane tricksters and wizards have exactly the same spell-casting text:

'Intelligence is your spell-casting ability for your wizard spells, since you learn your spells through dedicated study and memorization.'

This seems to imply that they cast their spells in the same mannner.

Presumably the answer to this would also apply to an eldritch knight/wizard multi-class as well?

Answer 1

Yes, as long as you are multiclassed

Since the spells gained from the Arcane Trickster's Spellcasting feature are referred to as "wizard spells", these spells would qualify for the wizard's Spellcasting Focus feature as long as you have it.

You know three 1st-level wizard spells of your choice ...  the Arcane Trickster Spellcasting table shows when you learn more wizard spells of 1st level or higher ... Whenever you gain a level in this class, you can replace one of the wizard spells you know with another spell of your choice from the wizard spell list. 

Spellcasting Focus [(Wizard)]

You can use an arcane focus as a spellcasting focus for your wizard spells.

(All emphasis mine)

Eldritch Knight?

Since the Eldritch Knight also uses similar language, the same rule applies.

Answer 2

As an Arcane Trickster, no. Once you multiclass, yes.

The wizard's Spellcasting feature has the following text:

Spellcasting Focus

You can use an arcane focus as a spellcasting focus for your wizard spells.

This section does not appear in the Arcane Trickster's Spellcasting feature, which would seem to imply that Arcane Trickster rogues cannot use an arcane focus; they're stuck with the more general rule that spellcasting needs a components pouch. However, the Arcane Trickster's Spellcasting feature always refers to the spells you know as "wizard spells":

You know three 1st-level wizard spells of your choice...

Arcane Trickster Spellcasting table shows when you learn more wizard spells of 1st level or higher...

Whenever you gain a level in this class, you can replace one of the wizard spells you know with another spell of your choice from the wizard spell list.

Knowing wizard spells does not automatically make you a wizard, or grant you access to any features of the wizard class. It's just there to explain that you use the wizard spell list, rather than the list for another class like cleric, or a specific Arcane Trickster list. This is true of Arcane Trickster, Eldritch Knight, and also the feat Magic Initiate when choosing wizard spells. However, the designation "wizard spells" does have two interesting effects, both relating to multiclassing into wizard. If an Arcane Trickster rogue were to take even one level of wizard, then all the spells granted by their Arcane Trickster Spellcasting feature would be eligible for any class features that mention "wizard spells". So, the wizard's Spellcasting Focus feature would now apply, allowing you to use an arcane focus for your Arcane Trickster spells.

Other effects of multiclassing like this include:

• Any class features that affect wizard spells can be applied to your Arcane Trickster spells. For example, a 14th-level wizard from the school of Evocation who also has at least three levels in rogue as an Arcane Trickster could apply their Overchannel feature to the spells they gained from Arcane Trickster.
• When you gain a level in rogue, you can replace "a wizard spell you know". Technically speaking, wizards don't "know" spells, they add them to their spellbook - the wording doesn't mention that you learn spells, and there's no "spells known" column on the Wizard table. However, if you have levels in Eldritch Knight, it uses the same wording as Arcane Trickster, so any spells from Eldritch Knight are "wizard spells that you know". This would imply that when you gain a level in Arcane Trickster, you can replace a spell gained from a level in Eldritch Knight, and vice versa. Admittedly, the multiclassing rules state that "You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class", but that's the general rule, and the wording for gaining a level in those specific classes talks about "wizard spells that you know", which one could argue is a more specific rule and thus overrides the general multiclassing rule. It's unclear, though, so that's probably something for the DM to rule until it's clarified by errata or Sage Advice.