Hi Jim,

Mmmm.  This is a tricky and rather sensitive problem.  You really need to
have a friendly chat with your neighbours about it.  Firstly, they may not
even know their cat is doing this.  Secondly, I doubt you'll put a stop to
it without their cooperation.

My neighbour called on me to say that my cat, Cheetah, had been entering
their house and eating their cat's food.  Naturally, I apologised then asked
for her cooperation in dealing with it.  We agreed that she could spray
water at my cat if Cheetah repeated the offence.  Further, that she could
catch Cheetah and deliver her to me for a good telling off.  After my
neighbour left, I had a better idea:  I picked up Cheetah and took her next
door.  Both the meighbour and I gave Cheetah a vigorous telling off right on
the threshold of 'the scene of crime'.  This worked stunningly well.
Cheetah never returned to the neighboour's house.

Your situation is more difficult because of the spraying in your home.  I
can only assume that the neighbour's cat wishes to extend his territory to
include your home, and to extend his female harem.  Clearly, this is not
acceptable.   It is essential to rid your home of his scent.  Certainly he
will want to come back in and refresh his previous marking and you will need
to prevent him, which is where it gets really tricky.

Have a friendly chat with your neighbours so they know you both have a
problem.  Sound them out as to what they would be prepared to do - and let
you do.
Perhaps you could suggest to your neighbours that they keep their male cat
in for a while.  Hopefully, his hormones will settle down so that he feels
less frisky and less dominant.  If this creates difficulties for your
neighbours, be prepared to compromise and offer to block your cat flat
temporarily, letting your cats out on request.  You may only be able to do
this at weekends if you're out at work, but it would still be a good start.
Try and get your neighbour's permission to spray their cat when it
trespasses, or to trap it and deliver it for a sound telling off.  Ask how
they admonish their cat when it's naughty; perhaps using their own technique
would have more effect.   My preferred option would be to have your
neighbour come in and severely admonish their cat at the 'scene of crime'.
It worked so sweetly for me!

All this relies, of course, on your neighbours being responsible cat owners.
We know they're responsible enough to doctor their cats, but they may, alas,
come from the "He does whatever he wants" school of cat training.
Goodness - this attitude makes me spit whiskers!

I truly hope you can resolve this.  Do let me know how you get on.
Spider

I Have A Cat Named Bootleg