Toyota Repair Problems? Ask a Mechanic for Answers ASAP
Can you tell me what you mean by "new" gearbox?
Cannot select with engine off, what does the shifter do?
Ok, sounds like something wrong with the reverse lockout mechanisim.
Did you have the shifter off prior to fitting the box?
A lot of manual transmissions won't allow reverse from 5th to prevent accidental reverse engagement while driving.
This is an internal lockout.
So if all forward gears are fine, it does the same with both gearsticks, and you can't select it with engine off, then there is an issue within the box.
Is reverse next to 1st or under 5th?
Unfortunately I would say you are pulling the transmission out, and disassembly as necessary to find out why.
Has someone had this trans apart recently?
Sounds like a shift rail interlock assembled incorrectly.
I sorry that I am not with you to help you decide.
But it sounds like an internal issue.
With the shifter off, you should be able to select all gears, so then you might see or feel an abnormality.
Otherwise the transmission needs to be disassembled.
Did you buy this trans with a warranty, or 2nd hand privately?
Do you need any further help?
I hope my advice has been of assistance to you.