Non-Jews in Synagogue

A Biblic Jewish synagogue must accept non-Jews in their synagogue.  The Bible makes this very clear in its description of how foreigners are to be treated.

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You also must love the foreigner, since you were foreigners in the land of Egypt.
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Deuteronomy 10:19

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“When a foreigner lives with you in your land, you must not oppress him.  You must regard the foreigner who lives with you as the native-born among you. You are to love him as yourself, for you were foreigners in the land of Egypt; I am Yehovah your God.
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Leviticus 19:33-34

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The assembly is to have the same statute for both you and the foreign resident as a permanent statute throughout your generations. You and the foreigner will be alike before the Lord. The same law and the same ordinance will apply to both you and the foreigner who resides with you.”
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Numbers 15:15-16

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“If a foreigner resides with you and wants to observe the Passover to the Lord, he is to do so according to the Passover statute and its ordinances. You are to apply the same statute to both the foreign resident and the native of the land.”
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Numbers 9:14

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Rejoice before Yehovah your God in the place where He chooses to have His name dwell—you, your son and daughter, your male and female slave, the Levite within your gates, as well as the foreigner, the fatherless, and the widow among you.
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Deuteronomy 16:11

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“Say to them: Anyone from the house of Israel or from the foreigners who live among them who offers a burnt offering or a sacrifice but does not bring it to the entrance to the tent of meeting to sacrifice it to the Lord, that person must be cut off from his people.
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Leviticus 17:8-9

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Moses wrote down this law and gave it to the priests, the sons of Levi, who carried the ark of the Lord’s covenant, and to all the elders of Israel. Moses commanded them, “At the end of every seven years, at the appointed time in the year of debt cancellation, during the Festival of Booths, when all Israel assembles in the presence of the Lord your God at the place He chooses, you are to read this law aloud before all Israel. Gather the people—men, women, children, and foreigners living within your gates—so that they may listen and learn to fear the Lord your God and be careful to follow all the words of this law. Then their children who do not know the law will listen and learn to fear the Lord your God as long as you live in the land you are crossing the Jordan to possess.”
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Deuteronomy 31:9-13

These quotes are from the Torah.  What they say is to accept foreigners (non-Jews), apply the same laws to them, rejoice before God with them, accept their sacrifices to God, and teach them the Torah.

When King Solomon built the first temple, he said:

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Even for the foreigner who is not of Your people Israel
but has come from a distant land
because of Your name—
for they will hear of Your great name,
mighty hand, and outstretched arm,
and will come and pray toward this temple—
may You hear in heaven, Your dwelling place,
and do according to all the foreigner asks You for.
Then all the people on earth will know Your name,
to fear You as Your people Israel do
and know that this temple I have built
is called by Your name.
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1 Kings 8:41-43

And in fact, to my knowledge, foreigners (non-Jews) with a genuine desire to worship God were never turned away from any of the three temples of Israel.  What does God say about these foreigners?

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No foreigner who has joined himself to the Lord
should say,
“The Lord will exclude me from His people”;
and the eunuch should not say,
“Look, I am a dried-up tree.”
For the Lord says this:
“For the eunuchs who keep My Sabbaths,
and choose what pleases Me,
and hold firmly to My covenant,
I will give them, in My house and within My walls,
a memorial and a name
better than sons and daughters.
I will give each of them an everlasting name
that will never be cut off.
And the foreigners who join themselves to the Lord
minister to Him, love the name of Yehovah
and become His servants,
all who keep the Sabbath without desecrating it
and who hold firmly to My covenant—
I will bring them to My holy mountain
and let them rejoice in My house of prayer.
Their burnt offerings and sacrifices
will be acceptable on My altar,
for My house will be called a house of prayer
for all nations.”
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Isaiah 56:3-7

The concept in the Bible is clear.  God wants Jews to welcome foreigners and allow foreigners to learn Torah.  In fact God clearly states the purpose of Judaism.

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“I, Yehovah, have called You
for a righteous purpose,
and I will hold You by Your hand.
I will keep You and appoint You
to be a covenant for the people
and a light to the nations,
in order to open blind eyes,
to bring out prisoners from the dungeon,
and those sitting in darkness from the prison house.
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Isaiah 42:6-7

How exactly can Judaism be "a light to the nations" if non-Jews are excluded from synagogues?

So why exactly are non-Jews excluded from rabbinic synagogues?  The answer is because the Romans commanded Jews not to convert others on penalty of death.  The rabbis, fearing the Romans, prohibited non-Jews from attending synagogue.  And as is typical of Rabbinic Judaism, tradition has become law for reasons long forgotten in spite of the fact that this law directly contradicts the spirit of the Torah.

In the world today, where can one find "a light to the nations"?  Only in Karaite synagogues because this is the only form of serious Judaism that accepts non-Jews.